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Statistics / A First Course in Probability 9 / Chapter 3 / Problem 70P

A First Course in Probability | 9th Edition | ISBN: 9780321794772 | Authors: Sheldon Ross

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There is a 50–50 chance that the queen carries the gene

Chapter 3, Problem 70P

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QUESTION:

Problem 70P

There is a 50–50 chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a 50–50 chance of having hemophilia. If the queen has had three princes without the disease, what is the probability that the queen is a carrier? If there is a fourth prince, what is the probability that he will have hemophilia?

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QUESTION:

Problem 70P

There is a 50–50 chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a 50–50 chance of having hemophilia. If the queen has had three princes without the disease, what is the probability that the queen is a carrier? If there is a fourth prince, what is the probability that he will have hemophilia?

ANSWER:

Step 1 of 2

Our goal is :

We need to find the probability that the queen is a carrier and

We need to find the probability he will have hemophilia.

Let ${Q}_{h}$ denotes that the queen carriers the gene for hemophilia and

Let ${Q}_{n}$ denotes that the queen does not.

Let ${P}_{i,h}$ denotes that the ${i}^{th}$ prince does not have hemophilia and

Let ${P}_{i,n}$ denotes that the event that he does not.

From the given information we know that.

$P({Q}_{h})=$ $\frac{1}{2}$ , $P({P}_{i,h}/{Q}_{h})=$ $\frac{1}{2}$ and $P({P}_{i,h}/{Q}_{n})=$ 0.

So $P({Q}_{n})$ is

$P({Q}_{n})=$ 1- $P({Q}_{h})$

$P({Q}_{n})=$ 1- $\frac{1}{2}$

$P({Q}_{n})=$ $\frac{1}{2}$

$P({P}_{i,n}/{Q}_{h})=$ 1- $P({P}_{i,h}/{Q}_{n})$

$P({P}_{i,n}/{Q}_{h})=$ 1- $\frac{1}{2}$

$P({P}_{i,n}/{Q}_{h})=$ $\frac{1}{2}$

Then $P({P}_{i,n}/{Q}_{n})$ is

$P({P}_{i,n}/{Q}_{n})=$ 1- $P({P}_{i,h})$

$P({P}_{i,n}/{Q}_{n})=$ 1

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