Recitation 9

Probability

1.

This is Problem 5.50 ``The probability of a royal flush''. A royal flush is the highest hand possible in poker. It consists of the ace, king, queen, jack and ten of the same suit. What is the probability of being dealt a royal flush in a five card deal? Solve this in stages using the same approach as in Exercise 5.48.

(a)

Start with spades.

i.

What is the probability that the first card is one that you need? (one of the ace, king, queen, jack or 10 of spades?)

ii.

What is the probability that the next card is another of the 4 that you need given that the first was one of the five that you needed?

iii.

What is the probability that the third card is one of the three that you need given that the first two were two of what you needed?

iv.

What is the probability that the fourth card is one of the two remaining cards you need given that you have three?

v.

What is the probability that the fifth card is the last remaining card you need given that you have 4 of them?

(b)

Now multiply the above 5 probabilities together to get the probability of a royal flush in spades.

(c)

Now multiply the result above by 4 to get the probability of a royal flush in any one of the 4 suits.

Confidence Intervals

2.

A 95% confidence interval for the mean $\mu$ of a population is computed from a random sample and found to be $9 \pm 3$. Circle the correct conclusion(s):

A:

there is a 95% probability that the population mean $\mu$ is between 6 and 12.

B:

there is a 95% probability that the true mean is 9 and the margin of error is 3.

C:

if we took many, many additional random samples and from each computed a 95% confidence interval for the population mean $\mu$ then approximately 95% of these intervals would contain the true population mean $\mu$.

3.

I calculate a 95% confidence interval from a sample of n=30 observations.

(a)

If I increase the sample size to n=100 would my interval be likely to be larger or smaller than for n=30?

(b)

If I change the level of confidence to 99% would my interval be larger or smaller than for 95%?

(c)

If I repeat the experiment and take 20 samples of size n=30 and calculate 20 independent 95% confidence intervals what is the distribution of the number of confidence intervals that contain the population mean $\mu$?