Probability

To solve the problem, we need to find the probability that a '5' will come up at least once when a fair six-sided die is thrown twice. We can approach this problem using the complementary probability method.

Step 1: Understand the total outcomes.
When a die is thrown twice, there are possible outcomes, since each throw has 6 possible results.

Step 2: Calculate the probability of not getting a '5' in a single throw.
The probability of not getting a '5' in a single throw is:

Step 3: Calculate the probability of not getting a '5' in both throws.
If the die is thrown twice and we do not get a '5' in either throw, the probability is the product of the probabilities of not getting a '5' in each throw (since the throws are independent):

Step 4: Use the complementary probability.
The probability of getting at least one '5' is the complement of the probability of not getting a '5' in both throws. Therefore:

Final Answer:
The probability that a '5' will come up at least once is .

To solve the problem, we need to determine the probability of getting at least two heads when three coins are tossed simultaneously. Let us break this down step by step.

Step 1: Understand the Sample Space When three coins are tossed, each coin has two possible outcomes: Heads (H) or Tails (T). Therefore, the total number of possible outcomes is: The sample space consists of all possible combinations of H and T for the three coins: Step 2: Identify Favorable Outcomes We are interested in the event of getting "at least two heads." This means we need to count the outcomes that have either exactly two heads or exactly three heads.
Case 1: Exactly Two Heads The outcomes with exactly two heads are: There are 3 such outcomes.
Case 2: Exactly Three Heads The outcome with exactly three heads is: There is 1 such outcome. Step 3: Count Total Favorable Outcomes The total number of favorable outcomes is the sum of the outcomes from both cases: Step 4: Calculate the Probability The probability of an event is given by the ratio of the number of favorable outcomes to the total number of possible outcomes:

Final Answer:

To solve this problem, we need to find the probability that a randomly chosen two-digit number is a multiple of 3.

1. Total Number of Two-Digit Numbers:
Two-digit numbers range from 10 to 99.
So, the total number of two-digit numbers is:

2. Counting Multiples of 3 Among Two-Digit Numbers:
The smallest two-digit multiple of 3 is 12, and the largest is 99.
We form an arithmetic sequence: 12, 15, 18, ..., 99 with common difference 3.
Using the formula for the nth term: So, there are 30 two-digit multiples of 3.

3. Calculating the Probability:

Final Answer:
The correct answer is (B) .

To solve this problem, we need to apply the basic rule of probability regarding complementary events.

1. Understanding Complementary Events:
If is the probability of an event , then is the probability of the event not happening.
The sum of the probabilities of an event and its complement is always 1:

2. Applying the Rule:
We directly apply the above rule to this question:

Final Answer:
The correct option is (C) 1.

To solve the problem, we need to find the probability of getting a red ball when one ball is drawn at random from a bag containing 4 black balls and 6 red balls.

1. Understanding Total Outcomes:
The total number of balls = 4 (black) + 6 (red) = 10
So, the total number of possible outcomes = 10

2. Favourable Outcomes:
The favourable outcomes (getting a red ball) = 6

3. Apply the Probability Formula:

Final Answer:
The probability of getting a red ball is .

To solve the problem, we need to identify the correct statement about the probability of an event .

1. Understanding the Range of Probability:
The probability of any event always lies between 0 and 1, inclusive. That is:

2. Evaluating the Given Options:
(1) → Incorrect (Probability can't be negative)
(2) → Incorrect (Probability can't be greater than 1)
(3) → Correct (This is the definition of probability)
(4) None → Incorrect (Option 3 is correct)

Final Answer:
The correct statement is .

To solve the problem, we need to find the probability of getting the same odd number on both dice when two dice are rolled.

1. Total Outcomes When Two Dice Are Rolled:
Each die has 6 faces, so the total number of outcomes when rolling two dice is:

2. Favorable Outcomes:
Odd numbers on a die are: 1, 3, 5
To get the same odd number on both dice, the favorable outcomes are:
(1,1), (3,3), (5,5)
So, the number of favorable outcomes = 3

3. Calculating the Probability:
Probability = (Number of favorable outcomes) / (Total outcomes)

Final Answer:
The probability of getting the same odd number on both dice is

To solve the problem, we need to find the probability of the complement of event , denoted as , given that .

1. Understanding the Complement Rule:
In probability, the sum of the probabilities of an event and its complement is always 1. That is:

2. Substituting the Given Value:
Given , we substitute into the equation:

3. Solving for :

Final Answer:
The value of is .

To solve the problem, we need to find the probability of not getting a red marble when a marble is drawn at random from a box containing 7 red marbles and 9 green marbles.

1. Total Number of Marbles:
The total number of marbles = 7 (red) + 9 (green) = 16 marbles

2. Favourable Outcomes:
The favourable outcome for the event "not getting a red marble" is drawing a green marble.
Number of green marbles = 9

3. Probability Formula:

Probability =

So, Probability =

Final Answer:
The probability of not getting a red marble is