Question

Suppose that a card is picked at random from 4 cards labeled as 1,2,....5 and the number that appears is x. A second card is picked at random from cards numbered 1,2,...x and its number is y. What is the conditional probability that x = 4 given y = 1?

Answer 1

The conditional probability that x = 4 given y = 1 is found by dividing the probability that x is 4 and y is 1 by the probability that y is 1. After computing the probabilities, the conditional probability is 3/25.

Step-by-step explanation:

To find the conditional probability that x = 4 given y = 1, we can apply the definition of conditional probability, which is P(A|B) = P(A and B)/P(B), where P(A|B) is the probability of A given B has occurred.

In this case, we want to find P(x=4|y=1). Now, if y=1, x can be 1, 2, 3, or 4 since y is selected from the range 1 to x. There is only one favorable scenario for x=4 out of these options. Therefore, the probability P(x=4 and y=1) is 1/4 (as there's only one way to pick y=1 from the numbers 1 to 4).

Moreover, P(y=1) is the sum of the probabilities of picking y=1 when x is 1, 2, 3, or 4. This means P(y=1) = 1/1 + 1/2 + 1/3 + 1/4 = 1 + 1/2 + 1/3 + 1/4. Simplifying that sum gives us 25/12.

Using the conditional probability formula, we get P(x=4|y=1) = P(x=4 and y=1) / P(y=1) = (1/4) / (25/12) = (1/4) * (12/25) = 3/25.

Thus, the conditional probability that x = 4 given that y = 1 is 3/25.