Conditional Probability

Given:

• and form a partition of the sample space . This means and are disjoint, and .

We want to find . We can use Bayes' theorem:

We know . We need to find and .

Since and form a partition:

We know and , so:

Also, we are given . Using the definition of conditional probability:

Now we have:

Finally, we can calculate :

Let the events be defined as follows:
Let represent the event that the car came from plant X.
Let represent the event that the car came from plant Y.
Let represent the event that the car is of standard quality.
We are given the following probabilities:
,

(80% of cars from plant X are of standard quality)
(90% of cars from plant Y are of standard quality)
We need to find , the probability that the car came from plant X given that it is of standard quality. By Bayes' Theorem: We first need to calculate , the total probability that the car is of standard quality: Substitute the given values: Now, using Bayes' Theorem: Thus, the probability that the car came from plant X is .

The correct answer is (C) : .

When two dice are thrown, the possible sums are the results of adding the numbers on each die.
We are told that the sum of the numbers on the dice is less than 6.
Therefore, we are interested in the cases where the sum of the dice is 2, 3, 4, or 5.
The possible outcomes for these sums are: - Sum = 2: (1, 1)
Sum = 3: (1, 2), (2, 1)
Sum = 4: (1, 3), (2, 2), (3, 1)
Sum = 5: (1, 4), (2, 3), (3, 2), (4, 1)
Thus, there are 10 possible outcomes where the sum of the dice is less than 6. Out of these, the favorable outcomes for a sum of 3 are (1, 2) and (2, 1), which are 2 outcomes. Therefore, the probability of getting a sum of 3, given that the sum is less than 6, is:

So, the correct answer is (C) : .

We are given the following information: From the conditional probability formula: Substituting the given values: Solving for : Next, we use the formula for the union of two events: Substituting the given values: Multiplying the entire equation by 10 to clear the denominators: Simplifying:

So, the correct answer is (B) : .

The correct answer is: (D):

We are given the following probabilities:

We are tasked with finding .

Step 1: Use the definition of conditional probability
Recall the definition of conditional probability:

Substitute into the equation:

From this, we can solve for in terms of :

Step 2: Use the definition of
Similarly, from the definition of conditional probability for , we have:

Substitute and into the equation:

This simplifies to:

Step 3: Solve for
From Step 1, we know that . Now substitute into this equation:

Solving for , we get:

Conclusion:
The probability of event is so the correct answer is (D): .