A previous year, the weights of the members of the San Francisco 49ers and the Dallas Cowboys were published in the San Jose Mercury News. The factual data are compiled into Table 3.24.
| Shirt# | ≤ 210 | 211–250 | 251–290 | 290≤ |
|---|---|---|---|---|
| 1–33 | 21 | 5 | 0 | 0 |
| 34–66 | 6 | 18 | 7 | 4 |
| 66–99 | 6 | 12 | 22 | 5 |
For the following, suppose that you randomly select one player from the 49ers or Cowboys.
If having a shirt number from one to 33 and weighing at most 210 pounds were independent events, then what should be true about P(Shirt# 1–33|≤ 210 pounds)?
The probability that a male develops some form of cancer in his lifetime is 0.4567. The probability that a male has at least one false positive test result (meaning the test comes back for cancer when the man does not have it) is 0.51. Some of the following questions do not have enough information for you to answer them. Write “not enough information” for those answers. Let C = a man develops cancer in his lifetime and P = man has at least one false positive.
- P(C) = ______
- P(P|C) = ______
- P(P|C') = ______
- If a test comes up positive, based upon numerical values, can you assume that man has cancer? Justify numerically and explain why or why not.
Given events G and H: P(G) = 0.43; P(H) = 0.26; P(H AND G) = 0.14
- Find P(H OR G).
- Find the probability of the complement of event (H AND G).
- Find the probability of the complement of event (H OR G).
Given events J and K: P(J) = 0.18; P(K) = 0.37; P(J OR K) = 0.45
- Find P(J AND K).
- Find the probability of the complement of event (J AND K).
- Find the probability of the complement of event (J OR K).
Use the following information to answer the next two exercises. Suppose that you have eight cards. Five are green and three are yellow. The cards are well shuffled.
Suppose that you randomly draw two cards, one at a time, with replacement.
Let G1 = first card is green
Let G2 = second card is green
- Draw a tree diagram of the situation.
- Find P(G1 AND G2).
- Find P(at least one green).
- Find P(G2|G1).
- Are G2 and G1 independent events? Explain why or why not.
Suppose that you randomly draw two cards, one at a time, without replacement.
G1 = first card is green
G2 = second card is green
- Draw a tree diagram of the situation.
- Find P(G1 AND G2).
- Find P(at least one green).
- Find P(G2|G1).
- Are G2 and G1 independent events? Explain why or why not.
Use the following information to answer the next two exercises. The percent of licensed U.S. drivers (from a recent year) that are female is 48.60. Of the females, 5.03% are age 19 and under; 81.36% are age 20–64; 13.61% are age 65 or over. Of the licensed U.S. male drivers, 5.04% are age 19 and under; 81.43% are age 20–64; 13.53% are age 65 or over.
Complete the following.
- Construct a table or a tree diagram of the situation.
- Find P(driver is female).
- Find P(driver is age 65 or over|driver is female).
- Find P(driver is age 65 or over AND female).
- In words, explain the difference between the probabilities in part c and part d.
- Find P(driver is age 65 or over).
- Are being age 65 or over and being female mutually exclusive events? How do you know?
Suppose that 10,000 U.S. licensed drivers are randomly selected.
- How many would you expect to be male?
- Using the table or tree diagram, construct a contingency table of gender versus age group.
- Using the contingency table, find the probability that out of the age 20–64 group, a randomly selected driver is female.
Approximately 86.5% of Americans commute to work by car, truck, or van. Out of that group, 84.6% drive alone and 15.4% drive in a carpool. Approximately 3.9% walk to work and approximately 5.3% take public transportation.
- Construct a table or a tree diagram of the situation. Include a branch for all other modes of transportation to work.
- Assuming that the walkers walk alone, what percent of all commuters travel alone to work?
- Suppose that 1,000 workers are randomly selected. How many would you expect to travel alone to work?
- Suppose that 1,000 workers are randomly selected. How many would you expect to drive in a carpool?
When the Euro coin was introduced in 2002, two math professors had their statistics students test whether the Belgian one Euro coin was a fair coin. They spun the coin rather than tossing it and found that out of 250 spins, 140 showed a head (event H) while 110 showed a tail (event T). On that basis, they claimed that it is not a fair coin.
- Based on the given data, find P(H) and P(T).
- Use a tree to find the probabilities of each possible outcome for the experiment of tossing the coin twice.
- Use the tree to find the probability of obtaining exactly one head in two tosses of the coin.
- Use the tree to find the probability of obtaining at least one head.
Use the following information to answer the next two exercises. The following are real data from Santa Clara County, CA. As of a certain time, there had been a total of 3,059 documented cases of AIDS in the county. They were grouped into the following categories:
| Homosexual/Bisexual | IV Drug User* | Heterosexual Contact | Other | Totals | |
|---|---|---|---|---|---|
| Female | 0 | 70 | 136 | 49 | |
| Male | 2,146 | 463 | 60 | 135 | |
| Totals |
Suppose a person with AIDS in Santa Clara County is randomly selected.
- Find P(Person is female).
- Find P(Person has a risk factor heterosexual contact).
- Find P(Person is female OR has a risk factor of IV drug user).
- Find P(Person is female AND has a risk factor of homosexual/bisexual).
- Find P(Person is male AND has a risk factor of IV drug user).
- Find P(Person is female GIVEN person got the disease from heterosexual contact).
- Construct a Venn diagram. Make one group females and the other group heterosexual contact.
Answer these questions using probability rules. Do NOT use the contingency table. Three thousand fifty-nine cases of AIDS had been reported in Santa Clara County, CA, through a certain date. Those cases will be our population. Of those cases, 6.4% obtained the disease through heterosexual contact and 7.4% are female. Out of the females with the disease, 53.3% got the disease from heterosexual contact.
- Find P(Person is female).
- Find P(Person obtained the disease through heterosexual contact).
- Find P(Person is female GIVEN person got the disease from heterosexual contact)
- Construct a Venn diagram representing this situation. Make one group females and the other group heterosexual contact. Fill in all values as probabilities.