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Chapter 1: Problem 31

What is the conditional probability that the first die is six given that the sum of the dice is seven?

Short Answer

Expert verified
The conditional probability that the first die is a six given that the sum of the dice is seven is \(\frac{1}{6}\).

Step by step solution

01

Determine the sample space for rolling two dice

When rolling two six-sided dice, there are a total of six possible outcomes for each die. This results in a sample space of size 6 x 6 = 36 possible outcomes (ordered pairs).
02

List the favourable outcomes for event A

Event A is the event that the first die shows a six. We can list these outcomes as follows: \((6,1), (6,2), (6,3), (6,4), (6,5), (6,6)\). There are 6 favourable outcomes for event A.
03

List the favourable outcomes for event B

Event B is the event that the sum of the dice is seven. We can list these outcomes as follows: \((1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\). There are 6 favourable outcomes for event B.
04

Find the favourable outcomes for both A and B events

To find the favourable outcomes for both events A and B, we look for outcomes that are in both event A and event B. This results in the outcome \((6,1)\), which is the only outcome where the first die is a six and the sum of the dice is seven.
05

Calculate the conditional probability P(A|B)

Now we can apply the conditional probability formula: P(A|B) = P(A and B) / P(B). First, we find P(A and B), which is the probability of the favourable outcome \((6,1)\). Since there is only one favourable outcome and there are 36 possible outcomes in total, P(A and B) = 1/36. Next, we find P(B), which is the probability of the event B (the sum of the dice is seven). There are 6 favourable outcomes for event B and 36 possible outcomes in total, so P(B) = 6/36 = 1/6. Finally, we calculate P(A|B) = P(A and B) / P(B) = (1/36) / (1/6) = (1/36) * (6/1) = 1/6. Thus, the conditional probability that the first die is a six given that the sum of the dice is seven is 1/6.

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Most popular questions from this chapter

In a certain species of rats, black dominates over brown. Suppose that a black rat with two black parents has a brown sibling. (a) What is the probability that this rat is a pure black rat (as opposed to being a hybrid with one black and one brown gene)? (b) Suppose that when the black rat is mated with a brown rat, all five of their offspring are black. Now, what is the probability that the rat is a pure black rat?

Use Exercise 15 to show that \(P(E \cup F)=P(E)+P(F)-P(E F)\).

Three dice are thrown. What is the probability the same number appears on exactly two of the three dice?

Sixty percent of the families in a certain community own their own car, thirty percent own their own home, and twenty percent own both their own car and their own home. If a family is randomly chosen, what is the probability that this family owns a car or a house but not both?

A fair coin is continually flipped. What is the probability that the first four flips are (a) \(H, H, H, H\) ? (b) \(T, H, H, H ?\) (c) What is the probability that the pattern \(T, H, H, H\) occurs before the pattern \(\mathrm{H}, \mathrm{H}, \mathrm{H}, \mathrm{H}\) ?
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