Practice Assignment (to be discussed randomly Oct. 8)

1. Suppose a friend reports that she has just had a string of "bad luck" with her car. She had three major problems in as many months and now has replaced many of the worn parts with new ones. She concludes that it is her turn to be lucky and she shouldn't have anymore problems for a while. Is she making the gambler's fallacy? Explain.

2. Explain what it means, statistically, to say there is a 30% chance of rain tomorrow.

3. 4.12 (p. 301)

4. Explain what is wrong with the following argument:

When two balanced dice are rolled, the sum of the dice can be 2,3,4,5,6,7,8,9,10,11, or 12. This gives 11 possibilities. Therefore the probability that the sum is 12 is 1/11.

What "rule" is being misapplied here?

5. Although it's not quite true, suppose the probability of having a male child (M) is equal to the probability of having a female child (F). A couple has four children.

(a) Are they more likely to have FFFF or to have MFFM? Explain your answer.

(b) Explain which sequence in part (a) a belief in the law of small numbers would lead people to say had higher probability?

(c) Is a couple with four children more likely to have four girls or to have two children of each sex? Explain. (Note in the next question you will reason it out probabilistically).

(d) Let X=number of girls in four children. Define the sample space for X. Find the probability for each value of 8. Hint: Start with the sample space for four children, e.g. FFFF, FFFM, and so on.

6. The Birthday Problem

How many people would need to be gathered together to be at least 50% sure that two of them share the same birthday?

(a) In a nonleap year (365 days) what is the probability that a second person does not have the same birthday as the first person.

Hint: There are 364 nonmatching birthdays and we can assume the 365 days are equally likely.

(b) What's the probability that the second person has a different birthday and a third person has a different birthday.

(c) In a group of 10 people, find the probability that at least two people share the same birthday.

Hint: Find the probability of the complement

(d) Try different values of n to see how many people we need to have probability > .5 that at least two share the same birthday.

(e) What is the probability with 50 people?

(f) A guest on the Tonight Show tried to explain the birthday problem to Johnny Carson, who could not believe the answer was 23. He polled the studio audience of 200 people, looking for a match for his own birthday, and found none. How did the guest on the Tonight Show mess up? That is, given the way the question was asked by Johnny, we aren't we surprised that there wasn't a match?

7. The probability that a randomly selected American adult belongs to the American Automobile Association (AAA) is .10 (10%) and the probability that they belong to the American Retired Persons (AARP) is .11 (11%). What assumption would we have to make in order to conclude that the probability that a person belongs to both is (.10)(.11)=.011? Do you think that assumption holds in this case? Explain.

8. In the "3 Spot" version of the California Keno lottery game, the player picks three numbers from 1 to 40. Ten possible winning numbers are then randomly selected. It costs $1.00 to play. The table below shows the possible outcomes.

(a) Compute the expected value for this game.

(b) Interpret what it means.