Probability

Let , , and . We want to find the probability that exactly two of them solve the problem. This can happen in three ways: and solve, and solve, or and solve.

$ P(C') = 1 - P(C) = 1 - \frac{1}{3} = \frac{2}{3} P(B') = 1 - P(B) = 1 - \frac{1}{4} = \frac{3}{4} P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2} $

Let be the event of getting an odd number when a die is thrown.

Let be the event of getting an even number when a die is thrown.

The possible outcomes when a die is thrown are {1, 2, 3, 4, 5, 6}.

The odd numbers are {1, 3, 5}.

The even numbers are {2, 4, 6}.

The probability of getting an odd number is .

The probability of getting an even number is .

The die is thrown 10 times. We are looking for the probability that an odd number comes up at least once. It's easier to calculate the probability of the complement event (that no odd numbers come up, meaning only even numbers appear) and subtract it from 1.

The probability of getting an even number in one throw is .

The probability of getting even numbers in all 10 throws is .

The probability of getting at least one odd number is .

Thus, the probability that an odd number will come up at least one time is .

Given that A, B, and C are three mutually exclusive and exhaustive events of an experiment, such that .

We need to find .

Since A, B, and C are mutually exclusive and exhaustive, we have:

Let . Then, and .

Substituting these into the equation , we get:

Therefore, .

Thus, the correct option is (C) .

We are given that the events A, B, and C are independent, and . The probability that at least two of the events A, B, and C occur is equivalent to the probability that either exactly two events occur or all three events occur. 1. **Probability of exactly two events occurring: The probability of exactly two events occurring can be calculated using the binomial probability formula for combinations. There are three possible combinations of two events happening: 2. **Probability of all three events occurring: Since the events are independent, the probability of all three events occurring is: 3. Total probability: The total probability that at least two events occur is the sum of the probabilities of exactly two events occurring and all three events occurring: Simplifying: Thus, the correct answer is (C): .

We are given the following probabilities: The probability of the complement , i.e., the probability that neither A nor B occurs, is given by the formula: Using the formula for the union of two sets: Substituting the known values: Therefore:

We are given: We are asked to find the inverse of , i.e.: Using the identity: we get: Now suppose . Then: So: But this contradicts the identity property. So let's compute the inverse of directly. Let: The inverse of a 2x2 matrix is: Here, So: Thus, Correct answer: A

The number of elements in the Cartesian product is given by the product of the number of elements in set and the number of elements in set . So, we have:

Since , we know that .

To find the value of , we need to find the possible values of and that satisfy .

The possible pairs of that satisfy are and .

For both pairs, we have:

Therefore, .

The correct option is (A) 50.

We have the following information:

Since A and B are independent events, we know that .

Substituting the given values, we have:

Since A and B are independent, the probability of their intersection is simply the product of their individual probabilities:

Substituting the values again, we have:

Simplifying the equation, we get:

Therefore, the value of is . The correct option is (C) .

Let:
→ Probability of lockdown
→ No lockdown
→ Probability that pandemic is controlled in one month given lockdown
→ Probability that pandemic is controlled in one month given no lockdown

Using the law of total probability:
Correct Answer: 0.65

1. Understand the problem:

We need the probability that at least one odd number appears when a die is thrown 10 times.

2. Calculate the probability of the complement event:

The probability of no odd numbers (i.e., all even numbers) in 10 throws:

3. Compute the desired probability:

4. Match the result to the options:

The probability corresponds to option (C).

Correct Answer: (C)