1. A soft drink company holds a contest in which a prize may be revealed on the inside

of the bottle cap. The probability that each bottle cap reveals a prize is 0.2, and

winning is independent from one bottle to the next.

You buy three bottles.

Let X= number of winning bottles out of 3 tries

(a) What is the sample space of X (in other words-what are all the values the random variable X can take on?

(b) Find the probability that none of your bottles reveals a prize.

(c) Find the probability that you win at least one prize.

(d) Can X be considered a Binomial Random Variable? If so what are n and ?

2. The table below gives the probability distribution of

X = number of meals eaten yesterday by individuals in a large population.

(a) Find P(X = 3). (Hint: What must the sum of all the probabilities in the samples space equal?)

(b) Find the probability that a randomly selected individual ate an odd number of

meals yesterday (1 or 3 meals).

(c) Find the probability that a randomly selected individual ate more than 2 meals

yesterday.

(d) Find the cumulative probability P(X 2).

3. Again using the scenario above in question2, calculate the

1. Expected value of the random variable X (X = number of meals eaten yesterday by individuals in a large population)
2. Find the standard deviation of the random variable X (X = number of meals eaten yesterday by individuals in a large population)

4. Suppose a box contains 5 red balls and 10 blue balls. If seven balls are selected at random WITH replacement, what is the probability that three are red? How many red balls do we expect to see on average when we select 7 balls?

5. Suppose the prevalence of a certain disease is 0.3. If people come into a clinic at random what is the probability that 5 people will come to the clinic before you see your 1^st person with the disease? Assume people show up at the clinic independently.

6. A report in LA times suggested that 1 in 200 people carry the defective gene that causes inherited colon cancer. In a sample of 1000 people, how many do you expect to carry the defective gene?

7. 10 people are going to be vaccinated with a measles vaccine. The probability of getting measles after being vaccinated is .05. Assume this is the same for every person and that each person is independent from the other. We are interested in the number of people who get measles after vaccination.

a. Does this have the features of a Binomial experiment? Why?

b. Find the probability that no people get the measles after being vaccinated.

c. Find the probability that exactly 3 people get the measles after being vaccinated.

d. Find the probability that at least 1 person gets the measles after being vaccinated.

e. Check your answers (b-d) using excel.

f. A measles vaccine was used in a village in Mexico. The supervisor of the field workers who performed the vaccinations suspected that the field workers did not actually vaccinate the village but were loafing off because of the 58 children said to be vaccinated 5 became infected with measles. Find the probability of seeing these data or more unusual? In other words, find the probability of seeing 5 or more children getting infected after vaccination if chance alone were operating. (Use excel). Based on this answer, what would you tell the supervisor and why?