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Chapter 3: Problem 1

Two fair dice are rolled. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers?

Short Answer

Expert verified
The conditional probability that at least one die lands on 6 given that the dice land on different numbers is \(\frac{1}{3}\).

Step by step solution

01

Calculate Probability of Dice Landing on Different Numbers (P(B))

In order to find the probability of the dice landing on different numbers, we need to first determine the number of ways this can happen. There are a total of 36 possible outcomes when two dice are rolled, but 6 of them result in pairs (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). Thus, there are 36 - 6 = 30 ways the dice can land on different numbers. So, P(B) = number of ways for different numbers / total outcomes, which gives: P(B) = 30/36
02

Calculate Probability of At Least One Die Landing on 6 and Dice Landing on Different Numbers (P(A 鈭� B))

Now we need to find the probability that, among the outcomes where the dice landed on different numbers, at least one die shows a 6. There are two ways this can happen: either die 1 shows a 6 and die 2 shows any number other than 6, or vice versa. There are 5 such outcomes for each case: - (6,1), (6,2), (6,3), (6,4), (6,5) - (1,6), (2,6), (3,6), (4,6), (5,6) In total, there are 10 outcomes where at least one die shows a 6 and the dice show different numbers. The probability of this happening can be calculated as: P(A 鈭� B) = 10/36
03

Calculate the Conditional Probability (P(A | B))

Now we have all the information to find the conditional probability using the formula: P(A | B) = P(A 鈭� B) / P(B). Plugging the values obtained in the previous steps, we get: P(A | B) = (10/36) / (30/36) The probabilities in the numerator and denominator both have "36" in their denominators, which allows us to simplify the expression: P(A | B) = 10/30 Further simplifying the fraction, we get: P(A | B) = 1/3 Therefore, the conditional probability that at least one die lands on 6 given that the dice land on different numbers is 1/3.

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Most popular questions from this chapter

An urn cont?ins \(b\) black balls and \(r\) red balls. One of the balls is drawn at random, but when it is put back in the urn, \(c\) additional balls of the same color are put in with it. Now, suppose that we draw another ball. Show that the probability that the first ball was black, given that the second ball drawn was red, is \(b /(b+r+c)\).

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