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Probability and Binomial Experiments - Statistics Questions and Answers

1. A project conducted by the Australian Federal Office of Road Safety asked people many questions about their cars. One question was the reason that a person chooses a given car, and that data is in Table 1 ("Car preferences," 2013).

Find the probability a person chooses a car for each of the given reasons.

Safety Reliability Cost Performance Comfort Looks

84 62 26 34 57 37

Table 1

2. Eyeglassomatic manufactures eyeglasses for different retailers. They test to see how many defective lenses they made in a time period. Table 2 gives the defect and the number of defects.

a.) Find the probability of picking a lens that is scratched or flaked.

b.) Find the probability of picking a lens that is the wrong PD or was lost in lab.

c.) Find the probability of picking a lens that is not scratched.

d.) Find the probability of picking a lens that is not the wrong shape.

Defect Type Number of Defects

Chamfer Wrong 1596

Crazing, Cracks 1546

Flaked 1992

Lost in Lab 976

Right Shape - Too Large 1105

Right shape - Too Small 4613

Scratch 5865

Spots & Bubbles 1371

Spots & Bubbles (intern made) 976

Wrong Axis 1838

Wrong Height 1130

Wrong PD 1398

Wrong Shape 1485

Table 2

3. In the game of roulette, there is a wheel with spaces marked 0 through 36 and a space marked 00.

a.) Find the probability of winning if you pick the number 7 and it comes up on the wheel.

b.) Find the odds against winning if you pick the number 7.

c.) The casino will pay you $20 for every dollar you bet if your number comes up. How much profit is the casino making on the bet?

4. How many ways can six books be chosen and arranged alphabetically if there are ten books to choose from?

5. How many ways can you choose seven people from a group of twenty?

6. Suppose you have an experiment where you flip a coin three times. You then count the number of heads.

a.) State the random variable.

b.) Write the probability distribution for the number of heads.

c.) Draw a histogram for the number of heads.

d.) Find the mean number of heads.

e.) Find the variance for the number of heads.

f.) Find the standard deviation for the number of heads.

g.) Find the probability of having two or more number of heads.

h.) Is it unusual to have the three flips result in two heads? Why or why not?

7. A big-brand-name dishwasher, which costs $800.00, has a 20% chance of needing to be replaced in the first 2 years of purchase. A two-year extended warranty costs $112.10 on a dishwasher. What is the expected value of the extended warranty assuming it is replaced in the first 2 years?

8. Suppose a random variable, x, arises from a binomial experiment. If n = 6, and p = 0.30, find the following probabilities (it is acceptable to use some form of technology such as web applet, Excel, calculator, etc.).

a.) P(x = 1)

b.) P(x = 5)

c.) P(x = 3)

d.) P(x ≤ 3)

e.) P(x ≥ 5)

f.) P(x ≤ 4)

9. The proportion of brown M&M’s in a milk chocolate packet is approximately 14% (Madison,2013). Suppose a package of M&M’s typically contains 52 M&M’s.

a.) State the random variable.

b.) Is a binomial experiment? Why or why not? Find the probability that

c.) Six M&M’s are brown.

d.) Twenty-five M&M’s are brown.

e.) All of the M&M’s are brown.

f.) Would it be unusual for a package to have only brown M&M’s? If this were to happen, what would you think is the reason?

10. Approximately 10% of all people are left-handed. Consider a grouping of fifteen people.

a.) State the random variable.

b.) Write the probability distribution.

c.) Draw a histogram.

d.) Describe the shape of the histogram.

e.) Find the mean.

f.) Find the variance.

g.) Find the standard deviation