Probability

Step 1: Total Number of Outcomes.
When two dice are thrown, the total number of possible outcomes is: This is because each die has 6 faces, and both dice are independent.
Step 2: Favorable Outcomes.
The favorable outcomes are those where at least one die shows a 6. We can calculate this using the complementary approach, i.e., the probability of getting at least one 6 is equal to 1 minus the probability of getting no 6 on either die. - The probability of not getting a 6 on one die is . - The probability of not getting a 6 on either die (both dice showing numbers other than 6) is: Therefore, the probability of getting at least one 6 is:
Step 3: Conclusion.
Thus, the probability of getting at least one 6 when two dice are thrown is:

Step 1: Identify possible combinations.
The sum of two integers is even if both integers are either even or both are odd. In this case, we need to find the probability that both integers selected are odd.
Step 2: Total possible outcomes.
The total number of ways to select 2 integers from the set is:
Step 3: Favorable outcomes.
The number of odd integers in the set is 6 (i.e., ). The number of ways to select 2 odd integers from these 6 is:
Step 4: Calculate the probability.
The probability that both selected integers are odd is: Thus, the probability is or approximately 0.273.

Step 1: Formula for Probability of Union.
The formula for the probability of the union of two events and is:
Step 2: Substitute the Given Values.
We are given that , , and . Substituting these values into the formula:
Step 3: Solve for .
To solve for , first find the sum of and : Now substitute into the equation: Solving for :
Step 4: Check for Independence.
Two events and are independent if: We know that and . Therefore: Since , we conclude that and are independent.
Step 5: Conclusion.
Hence, the events and are independent.

We have a total of 52 cards. - Number of Kings in the deck = 4 - Number of Aces in the deck = 4

Step 1: Probability that the first card is King.

Step 2: Probability that the second card is also King. After one King is drawn, 3 Kings remain and 51 cards are left.

Step 3: Probability that the third card is Ace. Now, 50 cards remain (4 Aces still there).

Step 4: Multiply probabilities (independent successive events).

Final Answer:

Let be the probability that solves the problem. Let be the probability that solves the problem. Thus, (i) Probability that the problem is solved. The problem is solved if either solves, or solves, or both solve. (ii) Probability that only one of them solves the problem. Case 1: solves, does not: Case 2: solves, does not: So, Final Answer:

Step 1: Define the experiment.
A die is thrown three times. Each throw has possible outcomes. Total number of outcomes:

Step 2: Use complementary probability.
We want the probability of at least one odd number.

Step 3: Probability of no odd number.
Odd numbers on a die: (3 outcomes). Even numbers on a die: (3 outcomes). Probability of even in one throw = . For three throws, probability of all even =

Step 4: Final probability.

Final Answer:

Step 1: Total possible outcomes. When two dice are thrown, total outcomes = . Given condition: sum of numbers = 6.

Step 2: Outcomes where sum = 6. So, total favourable cases for sum = 6 → .

Step 3: Outcomes where at least one 4 appears (within sum = 6). So, number of favourable outcomes = .

Step 4: Conditional probability.

Final Answer:

Let the event be that the ball drawn is black and event be that the ball is drawn from Container-III. We need to find , the probability that the ball is from Container-III given that it is black. By Bayes' Theorem: Now, calculate the individual probabilities: 1. is the probability of drawing a black ball from Container-III. This is the ratio of black balls in Container-III to the total number of balls in Container-III: 2. is the probability that Container-III is chosen. Since there are 3 containers, the probability of choosing any one container is . 3. is the total probability of drawing a black ball from any container. This can be calculated by adding the probabilities of drawing a black ball from each container, weighted by the probability of choosing each container: Simplifying: Finding a common denominator: Now, we can substitute into Bayes' Theorem: Conclusion: The probability that the ball is drawn from Container-III given that the ball is black is

We are given that: - The total number of students is 500.
- The number of girls is 230.
- 10% of the 230 girls are studying in class XII.
The number of girls studying in class XII is: The total number of students in class XII is unknown, but we need the probability that a randomly chosen student is both in class XII and is a girl. The probability is given by: Conclusion: The probability that a randomly chosen student is of class XII and is a girl is

We are given the following probabilities: The conditional probability is given by the formula: Substituting the given values: Conclusion: The value of is

(i) To find , we use the formula for the probability of the union of two events: Since A and B are independent events, Substituting the given values: Now, substitute into the formula for : (ii) We already know that Conclusion: (i) The value of is (ii) The value of is

We are given that there are three children, and at least one child is a girl. We are asked to find the probability that all three children are girls, given this condition.

Step 1: Total possible outcomes.
Each child can either be a boy or a girl. Hence, for three children, the total number of possible outcomes is: The possible combinations of boys (B) and girls (G) are:

Step 2: Restricting to the condition.
We are told that at least one child is a girl. Therefore, we eliminate the outcome where there are no girls. This leaves us with the following 7 possible outcomes:

Step 3: Favorable outcomes.
We want the probability that all three children are girls, which corresponds to the outcome .

Step 4: Calculate the probability.
The number of favorable outcomes is 1 (i.e., ), and the total number of possible outcomes, given that at least one child is a girl, is 7. Hence, the probability is: Conclusion: The probability that all three children are girls, given that at least one child is a girl, is:

We know the formula for conditional probability:

Step 1: Substitute values.

Step 2: Simplify.

Final Answer:

Step 1: Understanding the Concept:
This problem involves conditional probability. The notation represents the probability of event B occurring given that event A has already occurred. The relationship between conditional probability, joint probability, and individual probability is defined by the multiplication rule of probability.
Step 2: Key Formula or Approach:
The formula for the conditional probability of B given A is: To find the probability of the intersection of A and B, , we can rearrange this formula: Step 3: Detailed Calculation:
We are given the following values:



Using the rearranged formula: Substitute the given values into the formula: Now, we perform the multiplication: Since there are a total of four decimal places in 0.18 and 0.12, the result will have four decimal places. Step 4: Final Answer:
The value of is 0.0216.

Step 1: Understanding the Concept:
The phrase "solving the question" implies that at least one of the three students solves it. It is easier to calculate the probability of the complementary event, which is that none of the students solve the question, and then subtract this from 1.
Step 2: Key Formula or Approach:
Let A, B, and C be the events that students A, B, and C solve the question, respectively. We are given:
, , .
The probability that the question is solved is .
Using the complement rule:
Assuming the events are independent, .
Step 3: Detailed Explanation or Calculation:
First, calculate the probabilities that each student fails to solve the question.
Probability that A fails: .
Probability that B fails: .
Probability that C fails: .
Next, calculate the probability that all three of them fail. Since their attempts are independent events, we multiply their individual probabilities of failure.
Simplify the fraction:
Finally, the probability that the question is solved (at least one person solves it) is 1 minus the probability that no one solves it.
Step 4: Final Answer:
The probability of solving the question is .

Step 1: Understanding the Concept:
This question requires the use of fundamental probability rules, specifically the formula for conditional probability and the addition rule for probabilities.
Step 2: Key Formula or Approach:
1. Conditional Probability:
2. Addition Rule:
Step 3: Detailed Explanation:
Part 1: Find and .
We are given .
From this, we can deduce:
Part 2: Find .
Using the conditional probability formula: .
We are given and we know .
Rearranging the formula to solve for :
Part 3: Find .
Now use the addition rule for probabilities:
Substitute the values we found:
To add/subtract these fractions, find a common denominator, which is 26.
Step 4: Final Answer:
The value of is .

Step 1: Understanding the Concept:
We need to prove a relationship between the probabilities of two events, given that they are mutually exclusive. Mutually exclusive events are events that cannot happen at the same time.
Step 2: Key Formula or Approach:
1. Use the definition of mutually exclusive events: If and are mutually exclusive, their intersection is the empty set, i.e., . 2. Use the axiom of probability for the empty set: . 3. Use the addition axiom for mutually exclusive events: . 4. Manipulate these facts to arrive at the desired equation.
Step 3: Detailed Explanation:
The proof is based on the axioms of probability and the definition of mutually exclusive events.
By definition, if events and are mutually exclusive, then they cannot occur simultaneously. This means their intersection is the null or empty event. The probability of the empty event is zero. One of the fundamental axioms of probability (the addition rule for mutually exclusive events) states that: This equation holds true precisely because the events are mutually exclusive.
Now, let's look at the equation we need to prove: We can start from the addition axiom for mutually exclusive events: We can add 0 to the right side of the equation without changing its value: Since we have already established that for mutually exclusive events, we can substitute this into the equation: This completes the proof.
Step 4: Final Answer:
Using the definition and axioms related to mutually exclusive events, we have proven that the given identity holds true.

Step 1: Understanding the Concept:
This problem can be solved using the Law of Total Probability. This law is used to find the probability of an event when it depends on two or more mutually exclusive and exhaustive events.
Step 2: Key Formula or Approach:
Let C be the event of completing the work on time.
Let S be the event that a strike happens.
Let S' be the event that a strike does not happen.
The Law of Total Probability states: Using the conditional probability formula , this becomes: Step 3: Detailed Explanation:
From the problem statement, we can extract the following probabilities:
- Probability of a strike: - Probability of no strike: - Probability of completing on time, given no strike: - Probability of completing on time, given a strike happened: Now, we apply the Law of Total Probability to find : Calculate the individual terms: Add them together: Step 4: Final Answer:
The probability of completing the construction work in due time is 0.488.

Step 1: Understanding the Concept:
This is a fundamental axiom of probability theory. The probability of any event must be a value between 0 and 1, inclusive. We need to prove this using the basic definition of probability.
Step 2: Key Formula or Approach:
The classical definition of the probability of an event E is: We will use the properties of the sets E (event) and S (sample space).
Step 3: Detailed Explanation:
Let S be the sample space, which is the set of all possible outcomes of an experiment.
Let E be an event, which is a subset of the sample space ( ).
Let n(S) be the total number of outcomes in S and n(E) be the number of outcomes in E.
Part 1: Proving
The number of outcomes in any event E, , cannot be negative. The minimum possible number of outcomes is zero (for an impossible event). So, .
The total number of outcomes in a sample space, , must be positive for a non-trivial experiment, so .
Therefore, the ratio must be non-negative: Part 2: Proving
Since the event E is a subset of the sample space S ( ), the number of elements in E cannot exceed the number of elements in S. So, .
Dividing both sides of this inequality by the positive number , we get: Part 3: Combining the results
From Part 1 and Part 2, we have and .
Combining these gives the final result: Step 4: Final Answer:
By using the definitions of an event and sample space, we have proven that the probability of any event E must lie in the range [0, 1].

Step 1: Understanding the Concept:
This problem asks for a conditional probability. We are given an event that has already occurred (the sum of the numbers is 6), and we need to find the probability of another event (the number 4 appears at least once) given this condition.
Step 2: Key Formula or Approach:
The formula for conditional probability is , where:
- is the event that the number 4 appears at least once.
- is the event that the sum of the numbers is 6.
In terms of outcomes, this can be calculated as .
Step 3: Detailed Explanation:
First, let's define the sample space for the given condition (Event B).
Event B: The sum of the numbers is 6.
The possible outcomes for two dice throws that sum to 6 are: The total number of outcomes for event B is .
Next, let's define Event A.
Event A: The number 4 appears at least once.
Now, we need to find the intersection of A and B ( ), which means we look for outcomes in B where the number 4 appears at least once.
From the set of outcomes for B, the ones that contain at least one 4 are: The number of outcomes in is .
Step 4: Final Answer:
Using the formula for conditional probability: So, the conditional probability that the number 4 appears at least once, given that the sum is 6, is .

Step 1: Understanding the Concept:
This question deals with probability. A leap year has 366 days. We need to find the number of weeks and the number of remaining days to determine the probability of having an extra Tuesday.
Step 2: Key Formula or Approach:
Probability is calculated as: Step 3: Detailed Explanation:
A leap year has 366 days.
First, we find the number of complete weeks in 366 days by dividing by 7: This means a leap year always contains 52 full weeks. Therefore, there are guaranteed to be 52 Tuesdays.
The occurrence of a 53rd Tuesday depends on the two remaining (odd) days.
These two consecutive days can form the following possible pairs:
(Sunday, Monday)
(Monday, Tuesday)
(Tuesday, Wednesday)
(Wednesday, Thursday)
(Thursday, Friday)
(Friday, Saturday)
(Saturday, Sunday)
There are 7 possible outcomes for the pair of odd days.
For there to be 53 Tuesdays, one of these two odd days must be a Tuesday. The favorable outcomes are the pairs that include a Tuesday:
(Monday, Tuesday)
(Tuesday, Wednesday)
There are 2 favorable outcomes.
Step 4: Final Answer:
The total number of possible outcomes is 7, and the number of favorable outcomes is 2.
Therefore, the probability of a leap year having 53 Tuesdays is:

Step 1: Understanding the Concept:
This problem can be solved using the Law of Total Probability. The event of the work being completed on time depends on whether there is a strike or not. We need to consider both scenarios (strike and no strike), calculate the probability of completion in each case, and then sum them up to get the overall probability of completion.
Step 2: Key Formula or Approach:
Let S be the event that a strike occurs.
Let S' be the event that there is no strike.
Let C be the event that the construction work is completed on time.
The Law of Total Probability states: where is the probability of completion given a strike, and is the probability of completion given no strike.
Step 3: Detailed Explanation:
From the problem statement, we have the following probabilities:
Probability of a strike, .
Probability of no strike, .
Probability of completion given no strike, .
Probability of completion given a strike, .
Now, we apply the Law of Total Probability: Substitute the given values into the formula: Step 4: Final Answer:
The probability of the construction work being completed on time is 0.488.

Step 1: Understanding the Concept:
Two events, E and F, are said to be independent if the occurrence of one event does not affect the probability of the other event occurring. Mathematically, two events are independent if and only if . We need to calculate these three probabilities and check if the condition holds.
Step 2: Key Formula or Approach:
1. Define the sample space S for throwing a single die.
2. Identify the outcomes for event E (a multiple of 3).
3. Identify the outcomes for event F (an even number).
4. Identify the outcomes for the intersection event (an even multiple of 3).
5. Calculate the probabilities , , and .
6. Check if .
Step 3: Detailed Explanation:
1. Sample Space:
When a die is thrown once, the sample space is S = \{1, 2, 3, 4, 5, 6\}. The total number of outcomes is .
2. Event E:
E is the event that the number is a multiple of 3. The outcomes are E = \{3, 6\}.
The number of favorable outcomes is .
The probability is .
3. Event F:
F is the event that the number is even. The outcomes are F = \{2, 4, 6\}.
The number of favorable outcomes is .
The probability is .
4. Intersection Event :
is the event that the number is both a multiple of 3 and even. The only outcome is .
The number of favorable outcomes is .
The probability is .
5. Check for Independence:
We need to check if .
Let's calculate the product of and : We see that , which is equal to .
Step 4: Final Answer:
Since , the events E and F are independent.

Step 1: Understanding the Concept:
This question asks for the conditional probability of event A given that event B has occurred, denoted as . This is a direct application of the definition of conditional probability.
Step 2: Key Formula or Approach:
The formula for the conditional probability of A given B is: where is the probability of both A and B occurring, and is the probability of B occurring, with the condition that .
Step 3: Detailed Explanation or Calculation:
We are given the following probabilities: Using the formula for conditional probability: The denominators (13) cancel out: Step 4: Final Answer:
The value of is .

Step 1: Understanding the Concept:
The equation to be proved is the general multiplication rule (or chain rule) for the probability of the intersection of three events. The problem states that the events are independent, which is a specific condition. We can prove the identity in two ways: by showing it holds true for any set of events, or by using the independence condition to show both sides are equal. We will prove the general rule, which is always true.
Step 2: Key Formula or Approach:
The proof relies on the definition of conditional probability: Step 3: Detailed Explanation:
We want to prove .
Let's start with the left-hand side (LHS), . We can group the first two events and write this as: Now, apply the definition of conditional probability, treating as a single event 'A' and as event 'B'. Next, we can expand the term using the same definition of conditional probability: Now, substitute the expression for from equation (2) into equation (1): This is the same as the right-hand side (RHS). The identity is true for any three events, whether they are independent or not.
Note on Independence: The problem states that X, Y, and Z are independent. If this condition is applied:
By definition of independence, . (LHS)
Also by independence, and .
Substituting these into the RHS gives: .
Since both sides simplify to the same expression under the condition of independence, the identity holds true for independent events as well.
Step 4: Final Answer:
By applying the definition of conditional probability twice, we have shown that the identity holds. Hence proved.

Step 1: Understanding the Concept:
This question deals with conditional probability for independent events. Two events are independent if the occurrence of one does not affect the probability of the other. The notation represents the conditional probability of event B occurring given that event A has already occurred.
Step 2: Key Formula or Approach:
The definition of independent events A and B is that .
The formula for conditional probability is .
For independent events, this simplifies directly to .
Step 3: Detailed Explanation:
Given that events A and B are independent. By the definition of independence, the probability of B occurring is not affected by whether A has occurred or not. Therefore, the probability of B given A is simply the probability of B. We are given that . So, Alternative Method (using formula): First, calculate using the independence rule: Now, use the conditional probability formula: Both methods yield the same result.
Step 4: Final Answer:
The value of is 0.4.

Step 1: Understanding the Concept:
This is a problem of classical probability. The probability of an event is the ratio of the number of outcomes favorable to the event to the total number of possible outcomes in the sample space.
Step 2: Key Formula or Approach:
Step 3: Detailed Explanation:
When two fair six-sided dice are thrown, each die has 6 possible outcomes. The total number of possible outcomes in the sample space is: We want to find the probability of getting a sum of eight. Let's list all the pairs of outcomes (die 1, die 2) that sum to 8: There are 5 such pairs. So, the number of favorable outcomes is 5.
Now, we can calculate the probability: Step 4: Final Answer:
The probability of getting the sum eight is .

Step 1: Understanding the Concept:
We need to find the probability of the event "not A and not B", which is denoted as . We can find this using De Morgan's laws and the rule for the probability of a complement.
Step 2: Key Formula or Approach:
1. De Morgan's Law: .
2. Probability of a complement: .
3. Addition rule for probability: .
Step 3: Detailed Explanation or Calculation:
We want to find .
Using De Morgan's Law, we can write this as:
Using the rule for complementary events:
Now, we need to find using the addition rule:
Substitute the given values:
To add these fractions, find a common denominator, which is 8:
Finally, substitute this value back to find :
Step 4: Final Answer:
The probability of P(not A and not B) is .

Step 1: Understanding the Concept:
We need to find the probability of a specific outcome (1 girl and 2 boys) when making independent selections from three different groups. This can happen in several mutually exclusive ways (e.g., girl from group 1, boys from groups 2 & 3; or boy from group 1, girl from group 2, boy from group 3, etc.). The total probability is the sum of the probabilities of these individual cases.
Step 2: Key Formula or Approach:
1. Determine the probability of selecting a girl (G) and a boy (B) from each group.
2. Identify all possible combinations of selections that result in one girl and two boys.
3. Calculate the probability for each combination by multiplying the probabilities of the independent selections.
4. Sum the probabilities of all combinations.
Step 3: Detailed Explanation or Calculation:
Let's denote the probabilities for each group:
- Group 1: 3 Girls, 1 Boy (Total 4). .
- Group 2: 2 Girls, 2 Boys (Total 4). .
- Group 3: 1 Girl, 3 Boys (Total 4). .
The three possible ways to get exactly one girl and two boys are:
- Case 1: Girl from Group 1, Boy from Group 2, Boy from Group 3 (G B B)

- Case 2: Boy from Group 1, Girl from Group 2, Boy from Group 3 (B G B)

- Case 3: Boy from Group 1, Boy from Group 2, Girl from Group 3 (B B G)

The total probability is the sum of the probabilities of these mutually exclusive cases:
Step 4: Final Answer:
The probability that the three selected children consist of one girl and two boys is .

Step 1: Understanding the Concept:
We need to find the conditional probability of event A given that event B has occurred, denoted as P(A/B). We are given that the events A and B are independent.
Step 2: Key Formula or Approach:
The definition of conditional probability is:
The definition of independent events is:
Step 3: Detailed Explanation or Calculation:
For independent events, the occurrence of one event does not affect the probability of the other. We can show this using the formulas.
Substitute the formula for independent events into the conditional probability formula:
Since , we can cancel from the numerator and denominator:
We are given that .
Therefore, .
Step 4: Final Answer:
The value of P(A/B) is 0.3.

Step 1: Understanding the Concept:
This problem involves calculating the probability of two independent events occurring together. The outcome of drawing a ball from the first bag does not affect the outcome of drawing a ball from the second bag. For independent events, the probability that both occur is the product of their individual probabilities.
Step 2: Key Formula or Approach:
Let A be the event of drawing a black ball from the first bag, and B be the event of drawing a black ball from the second bag.
The probability of an event is given by: Since the events are independent, the probability of both events occurring is:
Step 3: Detailed Explanation or Calculation:
First Bag:
Total number of balls = 4 white + 2 black = 6 balls.
Number of black balls = 2.
The probability of drawing a black ball from the first bag is .
Second Bag:
Total number of balls = 3 white + 5 black = 8 balls.
Number of black balls = 5.
The probability of drawing a black ball from the second bag is .
Combined Probability:
The probability of getting a black ball from both bags is the product of the individual probabilities.
Step 4: Final Answer:
The probability of getting both black balls is .

Step 1: Understanding the Concept:
This is a problem of conditional probability, specifically drawing without replacement. The outcome of the second draw depends on the outcome of the first draw because the first ball is not returned to the bag.
Step 2: Key Formula or Approach:
The probability of two dependent events A and B occurring in sequence is given by the multiplication rule of probability:

Where is the probability of the first event, and is the probability of the second event given that the first event has occurred.
Let A be the event that the first ball is white, and B be the event that the second ball is white.
Step 3: Detailed Explanation or Calculation:
First, identify the initial state of the bag:
Number of white balls = 10
Number of black balls = 5
Total number of balls = 10 + 5 = 15
Event A: The first ball drawn is white.
The probability of drawing a white ball first is: Event B|A: The second ball drawn is white, given the first was white.
After drawing one white ball without replacement, the state of the bag changes:
Number of white balls remaining = 10 - 1 = 9
Total number of balls remaining = 15 - 1 = 14
The probability of drawing a second white ball is: Now, we find the probability that both balls are white: Simplify the fractions: Reduce the fraction to its simplest form by dividing the numerator and denominator by their greatest common divisor, which is 6: Step 4: Final Answer:
The probability that both balls taken out are white is .

Step 1: Recall the formula for conditional probability
Step 2: Substitute the given values Given:
Step 3: Solve for
Step 4: Final answer Verification: We can verify using the multiplication rule: