1. Human blood is classified by type (O, A, B, or AB) and Rh factor (Rh positive or Rh negative). 84% of people in the United States have Rh positive blood. What is the probability that any two people chosen at random from the US population have the same Rh factor?
Solution For two people chosen at random there are four outcomes for Rh factor.
| Outcome | ++ | +- | -+ | -- |
|---|---|---|---|---|
| probability | (0.84)(0.84) | (0.84)(0.16) | (0.16)(0.84) | (0.16)(0.16) |
Thus
p(both Rh same) = p(++) + p(--) = (0.84)(0.84) + (0.16)(0.16) = 0.7312
2. An insurance company sells $100,000 life insurance policies with a one year term to 50 year old men for a premium of $300. If the insured dies within one year of the start of the policy, the insurance company has to pay $100,000 to the policy's beneficiary. If the probability of a 50 year old male dying within one year is 1/400, what is the expected return for the insurance company on the sale of one policy? If the company sells 1000 such policies in one year, what is the probability that the company loses money at the end of the year after paying out any amounts it has to pay out?
Solution The probability distribution for a single policy looks like this.
| Outcome | $300 | $300 - $100000 |
|---|---|---|
| probability | | |
The expected return on a single policy is the mean of this distribution.
If the company sells 1000 policies, they collect $300,000 in premiums at the start of the year. The company will lose money if more than 3 people die in that year. Let X be the count of policy holders who die in that year. X is governed by the binomial distribution and has a distribution B(1000,1/400). Since n is fairly large, we can approximate the binomial distribution with a normal distribution having
To compute 
we convert from X to z
and compute
3. The distribution of annual returns on common stocks is roughly symmetric, but extreme observations are more frequent than in a normal distribution. Because the distribution is not strongly nonnormal, the mean return over even a moderate number of years is close to normal. In the long run, annual real returns on common stocks have varied with a mean of about 9% and a standard deviation of about 28%. Suppose you plan to put $2000 into stocks today and withdraw the money 10 years from now. Assuming that stocks continue to perform as they have in the past, what is the probability that the mean annual return on your money is greater than 10%?
Solution Let 
be the average return over 10 years. Assuming that 
is governed by the normal distribution we have
The probability that the mean annual return exceeds 10% is the same as the probability that z exceeds
From the table
4. A survey of the pay of CEOs of 104 corporations showed that in the last year their average compensation increased by 6.9%. The numbers in the survey had a standard deviation of 55%. Give a 95% confidence level range for increase in compensation. Is this result sufficient evidence to conclude that the average pay for all CEOs went up last year? (Use a significance level of 5%).
Solution Assuming that the standard deviation for CEO pay is 55%, the 95% confidence interval for this result is
To test whether or not this is significant evidence that the average pay for all CEOs went up last year, we formulate a null hypothesis that says their pay stayed the same
and an alternative hypothesis that says the actual 
is greater than 0.
With that hypothesis, the observed value of 
has associated probability of
This is not significant at the 5% level.