Problem 47 - A First Course in Probability

Question

An urn contains 5 white and 10 black balls. A fair die is rolled and that number of balls is randomly chosen from the urn. What is the probability that all of the balls selected are white? What is the conditional probability that the die landed on 3 if all the balls selected are white?

Step-by-Step Solution

Answer

The probability of selecting all white balls is \[ P(\text{All white}) = \frac{C(5, n)}{C(15, n)} \times \frac{1}{6}, \] summed for all roll values \(n = 1, \dots, 6\). The conditional probability that the die landed on 3 if all the balls selected are white is \[ P(\text{Roll 3} | \text{All white}) = \frac{P(\text{All white} | \text{Roll }= 3) \times P(\text{Roll }= 3)}{P(\text{All white})}. \]

Step 1: Calculate the probability of selecting all white balls for each roll value

To do this, we'll calculate the probability of selecting only white balls for each possible roll value, i.e., n = 1, 2, 3, 4, 5, and 6. Since there are a total of \(15\) balls in the urn, with \(5\) being white and \(10\) being black, for each possible roll value, the probability of selecting all white balls can be calculated as a ratio of the number of ways of choosing only white balls to the number of ways of choosing any \(n\) balls. We can use the combination formula to calculate the number of ways of selecting balls: \[ C(n, k) = \frac{n!}{k!(n - k)!}, \] where \(n\) is the total number of items, \(k\) is the number of items to be selected, and ! denotes the factorial of a number.

Step 2: Calculate the weighted probability

Since there are 6 possible roll values and each has equal probability (\(\frac{1}{6}\)), we need to find the weighted probability of selecting all white balls based on each roll value. The probability for each roll value can be calculated as: \[ P(\text{All white} | \text{Roll }= n) = \frac{C(5, n)}{C(15, n)} \times \frac{1}{6}. \] Summing the probabilities for all 6 roll values will give us the total probability of selecting all white balls.

Step 3: Calculate the conditional probability

Now that we have the probabilities for each roll value, let's calculate the conditional probability that the die landed on 3 if all the balls selected are white. We can use the formula for conditional probability: \[ P(\text{Roll 3} | \text{All white}) = \frac{P(\text{All white} | \text{Roll }= 3) \times P(\text{Roll }= 3)}{P(\text{All white})}. \] We already have each of these probabilities except for \(P(\text{All white})\). We can calculate this by summing the probabilities for each roll value as calculated in step 2. Once we have all the probabilities, we plug them into the formula to get the conditional probability.