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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

(a) A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. What is the probability that it is the fair coin? (b) Suppose that he flips the same coin a second time and again it shows heads. Now what is the probability that it is the fair coin? (c) Suppose that he flips the same coin a third time and it shows tails. Now what is the probability that it is the fair coin?

Let \(E, F, G\) be three events. Find expressions for the events that of \(E, F, G\) (a) only \(F\) occurs, (b) both \(E\) and \(F\) but not \(G\) occurs, (c) at least one event occurs, (d) at least two events occur, (c) all three events occur, (f) none occurs, (g) at most one occurs, (h) at most two occur.

If two fair dice are tossed, what is the probability that the sum is \(i\), \(i=2,3, \ldots ., 12 ?\)

Suppose that 5 percent of men and \(0.25\) percent of women are colorblind. A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal number of males and females.

We say that \(E \subset F\) if every point in \(E\) is also in \(F\). Show that if \(E \subset F\), then $$ P(F)=P(E)+P\left(F E^{\varepsilon}\right) \geq P(E) $$
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