When Michael Jordan came up to try for a sixth free throw after having made five straight free throws, the announcer commented that the law of averages would be working against Jordan (Barron's AP Statistics Review, 1st Edition, p. 314, Question #6)
1. What did the announcer mean? Was this a correct interpretation
of probability? Explain.
2. Jordan makes 90% of his free throws.
a. What is the probability that he will make six straight free throws?
b. That he will make five straight free throws and then miss the next?
c. That he will make the next free thrown given that he has made the
last five in a row?
3. Describe how you could use a random number table to set up a simulation
to analyze the situation the announcer was commenting on.
4. If Jordan shoots six times from the free throw line, what are the mean and standard deviation for the number of shots he is expected to make?
5. If Jordan makes only 35 out of 40 free throws during the playoffs,
is this sufficient evidence that the probability of his making a free throw
is really below .90? Explain in excruciating detail.
6. Explain carefully how you would assign random digits to simulate
the outcome of the situations below.
a. P(rain) = .4
b. P(rain ) = .43
c. P(rain) = .377
7. On a certain day the blood bank needs 4 donors with type O blood. How many donors, on average, would they have to see to get exactly four donors with type O blood, assuming that 45% of the population has type O blood? (Use the random number table. Show which line you started at and mark appropriately.)
8. Suppose it took the blood bank 12 donors to find 4 with type O blood.
This surprised the director since it usually doesn’t take that many donors.
Could this have occurred by random chance or were people with type O blood
particularly stingy on that day? Use the results from problem 7 to
find the probability it takes 12 or more donors to get 4 with type O blood.
9. Last year I bought a string of 6 Christmas lights (6 lights was all
I could afford...). Unfortunately, my little string of 6 lights stopped
working after only 9 days. Did I get a defective string, or could
this have occurred by random chance? The manufacturer says that all
six light bulbs on a string stop working if one or more bulbs born out
and that the probability that any individual bulb burns out is 0.1.
a. Design and conduct a simulation (do 20 runs) to estimate the probability that a string of six bulbs will go out.
b. Calculate the theoretical probability that all bulbs will work. ___________
Calculate the theoretical probability that at least one bulb doesn’t work. __________
10. In many countries, it is desirable to have sons. Therefore, couples are inclined to keep having children until they get a son. Some countries, however, would like to discourage this practice. In an alternate universe, suppose the government of a country established a policy that says families are allowed to have children until they get a boy, and then they must stop, even if this is their only child. Design and conduct a simulation (do 20 runs) to answer the following, assuming that births are equally likely to result in boys and girls.
a. What is the average number of children per family?_____________ Average number of girls? ________
b. If this policy is practiced, will overpopulated countries continue to grow?
c. A certain couple was determined to have a son. This couple had produced three girls without a boy. How likely is it to take 4 or more children before having a boy?
d. What type of distribution does this simulation model? ________________
Use the appropriate cdf (cumulative density function) on your calculator
to find the exact theoretical probability for the question in part c.