Question

There are two boxes. The first box contains 4 balls, numbered from 1 to 4 . The second box contains 5 red balls and 8 blue balls. Bill selects a ball from the first box and then he selects, from the second box, a number of balls equal to the number on the ball selected from the first box. (a) What is the probability that Bill selects only red balls from the second box? (b) What is the probability that Bill selected ball number 2 from the first box, given that he selected only red balls from the second box?

Answer 1

To solve part A, calculate the probability for each possible draw from box one and combine them. For part B, apply Bayes' theorem, considering the probability of drawing two red balls from box two and the probability of having picked ball number 2 from box one.

Step-by-step explanation:

Probability of Selecting Red Balls

Part A: The probability that Bill selects only red balls from the second box can be found by considering each possible number he could select from the first box:

If Bill selects the ball numbered 4, the probability of four red balls in succession is (5/13) × (4/12) × (3/11) × (2/10).

To find the total probability, you must add up the probabilities of each case, each weighted by the probability of drawing the corresponding number from the first box (which is 1/4 for each number).

Conditional Probability

Part B: The probability that Bill selected ball number 2 from the first box given that he selected only red balls from the second box is found using Bayes' theorem. This theorem relates the conditional probability to the likelihood of the event occurring with the overall probability of selecting only red balls.

Using the probabilities we calculated in Part A, we can apply Bayes' theorem:

P(Ball 2 | Only Red Balls) = P(Only Red Balls | Ball 2) × P(Ball 2) / P(Only Red Balls)

Here, we plug in the probabilities:

P(Ball 2 | Only Red Balls) = (5/13) × (4/12) × 1/4 / P(Only Red Balls)

We previously calculated P(Only Red Balls) as the total probability of drawing only red balls from all cases in Part A.

Answer 2

The student's question requires calculating varying probabilities of selecting red balls from the second box based on the first box selection and using Bayes' theorem to determine the likelihood of selecting a specific numbered ball, given that only red balls are drawn from the second box.

Step-by-step explanation:

The problem involves calculating probabilities for a two-stage random selection process. When trying to ascertain the probability of selecting only red balls from the second box, we need to consider the various possibilities stemming from the first selection. There is a different probability of selecting red balls based on the number of balls Bill chooses to draw from the second box (determined by the number on the ball selected from the first box).

For part (a): We need to calculate the probability of selecting only red balls when 1, 2, 3, or 4 balls are drawn from the second box, and then take the weighted average based on the first box probabilities. The probability of selecting 1, 2, 3, or 4 red balls from the second box is calculated using combinations and taking into account the total number of red balls (5) against the total number of balls possible to choose from (13).

Part (b) requires us to use Bayes' theorem. Here, we are given the condition that Bill selected only red balls from the second box and we need to determine the probability that the number 2 ball was selected from the first box.