Probability

The probability of an event lies between 0 and 1, inclusive. This means: Option (A) 0 is a valid probability, representing an impossible event.
Option (B) 1 is a valid probability, representing a certain event.
Option (D) 0.99999 is a valid probability, as it is still within the range [0, 1].
However, option (C) 1.0001 is not a valid probability because it exceeds the upper limit of 1.

The correct option is (C):

Total number of pens in the box = 42 blue + 22 black = 64 pens.
Since the first pen picked was black, the total number of pens left in the box is 63, and the number of blue pens remaining is 42.
Thus, the probability of picking a blue pen as the second pen is:

The correct option is (B):

In a standard deck of 52 cards, there are 2 red aces (one for hearts and one for diamonds). The total number of cards in the deck is 52. So, the probability of drawing a red ace is given by: Thus, the correct answer is , but there seems to be an issue with the options provided in the image. Hence, it’s possible that option 1 might be considered the closest match. However, the exact probability should be .

The correct option is (C):

The total area of the rectangle is given by: The area of the circular region is: Thus, the winning probability is the ratio of the area of the circle to the area of the rectangle: Approximating as :

The correct answer is option (A):

When a die is thrown once, there are 6 possible outcomes, as the die has 6 faces numbered from 1 to 6. When the die is thrown twice, the total number of possible sample events can be found by multiplying the outcomes of the first throw by the outcomes of the second throw. Thus, the number of sample events is: Therefore, the number of sample events when a die is thrown twice is 36.

The correct option is (D):

Correct answer: 8

Explanation:
When a coin is tossed, it has 2 possible outcomes: Heads (H) or Tails (T). For three coins, each coin can independently show either Heads or Tails. Therefore, the total number of outcomes for three coins is: The possible outcomes are:

Hence, the total number of outcomes is 8.

Correct answer:

Explanation:
Two events and are said to be mutually exclusive if they cannot occur at the same time. This means that the intersection of and , denoted by , is the empty set, i.e., . Therefore, the value of is , which represents the empty set.

Hence, .

Correct answer:

Explanation:
The total number of balls in the bag is: The number of red balls is 6. Therefore, the probability of drawing a red ball is:

Hence, the probability of drawing a red ball is .

Correct answer:

Explanation:
In a non-leap year, there are 365 days. Dividing 365 by 7 gives: Therefore, there will be exactly 52 occurrences of each day of the week, plus one additional day. The extra day could be any of the seven days of the week, so there is a chance that the extra day will be a Thursday. If the extra day is a Thursday, there will be 53 Thursdays in that year. Hence, the probability that a non-leap year will have 53 Thursdays is .

Therefore, the probability is .

A deck of 52 cards has 26 red cards (13 diamonds and 13 hearts). Each suit has 3 face cards (Jack, Queen, and King). Hence, the total number of red face cards is: Thus, the probability of selecting a red face card is:

The correct option is (A):

The total number of balls in the box is: The number of balls that are not white is: Thus, the probability that a ball drawn at random is not white is:

The correct option is (C):

When two dice are thrown together, each die has 6 faces. The total number of possible outcomes is: The favorable outcomes for getting the same number on both dice are: There are 6 favorable outcomes. Therefore, the probability of getting the same number on both dice is:

The correct option is (C):

The probability of an event must always be a number between 0 and 1, inclusive. This means that: Now, let's evaluate each option:
is a valid probability because it lies between 0 and 1.
0.3 is a valid probability because it is between 0 and 1.
33% is equivalent to , which is a valid probability because it lies between 0 and 1.
is greater than 1, which is not a valid probability.

The correct option is (D):

• 0: Represents an impossible event.
• 1: Represents a certain event.
• : This is a fraction that lies between 0 and 1, representing a valid probability.

• : This equals 1.25, which is greater than 1. As probabilities cannot exceed 1, cannot represent a probability.

Given: A well-shuffled deck of 52 playing cards.

Step 1: Understanding Face and Non-Face Cards

- A deck has 52 cards.
- Face cards: Jacks, Queens, and Kings.
- There are 3 face cards per suit (Hearts, Diamonds, Clubs, Spades).
- Total face cards = .
- Non-face cards = .

Step 2: Finding the Probability

Final Answer:

Given: A bag contains 3 red balls and 5 black balls.

Step 1: Understanding the Probability of Selecting a Red Ball

Total balls = 3 + 5 = 8

Red balls = 3

Step 2: Probability Calculation

Probability of selecting a red ball = Number of red balls / Total number of balls

= 3 / 8

Final Answer: 3 / 8

Let:

be the number of correct answers.

be the number of wrong answers.

Step 1: Forming equations

Given that the total number of questions answered is:

...(1)

Marks are awarded as follows:

- Each correct answer gives mark.

- Each wrong answer results in a deduction of mark.

Total marks obtained:

...(2)

Step 2: Solve for and

From equation (1):

Substituting in equation (2):

Final Answer: 100

Initially, a standard deck of cards has 52 cards.

There are 13 cards of each suit: hearts, diamonds, clubs, and spades.

Hearts and diamonds are red, while clubs and spades are black.

So there are 26 red cards and 26 black cards.

We are given that 2 cards of hearts are missing and 4 cards of spades are missing.

The total number of missing cards is .

The number of cards remaining is .

Originally, there were 13 spades, but 4 are missing.

So the number of spades remaining is .

Originally, there were 13 clubs, and no clubs are missing.

So the number of clubs is 13. The total number of black cards remaining is .

The probability of getting a black card from the remaining pack is the number of black cards remaining divided by the total number of cards remaining: $ $

When two dice are thrown, there are a total of possible outcomes.

We want to find the number of outcomes where the sum of the two numbers is more than 10.

This means the sum is 11 or 12.

The outcomes that result in a sum of 11 are: (5, 6) and (6, 5)

The outcomes that result in a sum of 12 are: (6, 6) So, there are 2 outcomes with a sum of 11 and 1 outcome with a sum of 12. The total number of outcomes where the sum is more than 10 is .

Therefore, the probability that the sum of the two numbers is more than 10 is: $ \frac{1}{12}$.

We are given that $ x \frac{5}{8} x x$ is 4.5.

Let be the number of red balls, be the number of white balls, and be the number of blue balls.

We are given that , , and .

The total number of balls in the box is 24. So, .

Substituting the given values, we have

Thus, , , and .

We want to find the probability that the selected ball is not red.

This means the selected ball is either white or blue.

The number of balls that are not red is .

The probability of selecting a ball that is not red is the number of non-red balls divided by the total number of balls: $ \frac{5}{6}$.

Step 1: Understanding the Probability.
There are 365 days in a year (ignoring leap years), and each person has an equal chance of being born on any of these days. For the probability that Ram and Syam will have the same birthday, the first person (Ram) can have his birthday on any of the 365 days, but for Syam to have the same birthday, he must be born on the same day as Ram.
Step 2: Probability Calculation.
The probability that Syama's birthday matches Rama's birthday is: However, for both to have the same birthday, the event happens only after considering the first personas (Ram's) birthday choice, so the probability is for Syam.
Step 3: Conclusion.
Thus, the probability that both Ram and Syam will have the same birthday is:

Step 1: Understanding the Total Cards and Queen Cards.
A standard deck of cards consists of 52 cards. There are 4 queens in the deck, one from each suit (hearts, diamonds, clubs, and spades).
Step 2: Probability Formula.
The probability of an event is given by the formula: In this case, the number of favorable outcomes is the number of queen cards, which is 4, and the total number of outcomes is the total number of cards, which is 52.
Step 3: Substituting Values.
The probability of drawing a queen card is: Simplifying:
Step 4: Conclusion.
Thus, the probability of drawing a queen card from a deck of 52 cards is .

Step 1: Understand equally likely outcomes
Equally likely outcomes refer to situations where each possible outcome has the same probability of occurring.
Step 2: Analyze each option
Option (1): Tossing a coin has two equally likely outcomes: heads and tails.
Option (2): Tossing two coins simultaneously has four equally likely outcomes: (HH, HT, TH, TT).
Option (3): Rolling two dice also has 36 equally likely outcomes, as each die has 6 faces, giving 6 × 6 = 36 possibilities.
Step 3: Conclusion
All of the options represent situations with equally likely outcomes. Hence, the correct option is .

Step 1: Understand the relationship for complementary events
The probability of the complement of an event is given by: This represents the probability that event does not occur.
Step 2: Identify the correct option
The correct formula for the probability of the complement of event is .
Step 3: Conclusion
Therefore, the correct option is .

Step 1: Understand the concept of probability
Probability is a measure of the likelihood of an event occurring. The probability of an event is always a number between 0 and 1, inclusive. That is: A probability of 0 means the event cannot occur. A probability of 1 means the event will certainly occur.
Step 2: Analyze each option
Option (1): is a valid probability, as it is a number between 0 and 1.
Option (2): 15% is equivalent to 0.15, which is a valid probability.
Option (3): 0.7 is a valid probability, as it is between 0 and 1.
Option (4): -1.5 is not a valid probability because probabilities cannot be negative.
Step 3: Conclusion
The only option that cannot be a probability is -1.5, as it falls outside the valid range for probabilities.