# Find Expected Results and Probability

1. If the IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.

(a) Find the probability that a randomly selected person has an IQ score between 88 and 112. (Show work)

(b) If 100 people are randomly selected, find the probability that their mean IQ score is greater than 103. (Show work)

2.Imagine you are in a game show. There are 4 prizes hidden on a game board with 16 spaces. One prize is worth $4000, another is worth $1500, and two are worth $1000. You have to pay $50 to the host if your choice is not correct.

(a) What is your expected winning in this game? (Show work)

(b) If you are offered a sure prize of $400 in cash, and you can just take the money without playing the game. What would be your choice? Take the money and run, or play the game? Please explain your decision.

3. Mimi just started her tennis class three weeks ago. On average, she is able to return 25% of her opponent's serves. If her opponent serves 10 times, please answer the following questions:

(a) What is the probability that she returns at most 2 of the 10 serves from her opponent? (Show work)

(b) How many serves can she expect to return? (Hint : What is the expected value?) (Show work)

4. Men's heights are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. Mimi is designing a plane with a height that allows 95% of the men to stand straight without bending in the plane. What is the minimum height of the plane? (Show work)

5. There are 7 seniors and 3 juniors in the statistics club, and a team of 5 will be randomly selected to attend the Joint Statistical Meetings in San Diego.

(a) What is the probability that all 3 juniors are picked in this team of 5? (Show work)

(b) What is the probability that none of the juniors are picked in this team of 5? (Show work)

(c) What is the probability that 2 of the juniors are picked in this team of 5? (Show work)

6. A soda company wants to stimulate sales in this economic climate by giving customers a chance to win a small prize for every bottle of soda they buy. There is a 15% chance that a customer will find a picture of a cherry at the bottom of the cap upon opening up a bottle of soda. The customer can then redeem that bottle cap with a picture of a cherry for a small prize. Now, if I buy a 6-pack of soda, what is the probability that I will win something, i.e., at least win a single small prize? (Show work)

7. A telemarketing company has two customer service teams. Team A has 20 agents and Team B has 30 agents. 10 agents in Team A and 20 agents in Team B contact customers via e-mails, and the rest contact customers via phone. Find the probability of getting someone who is from Team A, given that the selected person uses phone to contact customers. (Show work)

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#### Solution Preview

1. If the IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.

(a) Find the probability that a randomly selected person has an IQ score between 88 and 112. (Show work)

z1=(88-100)/15=-0.8

Z2=(112-100)/15=0.8

P(-0.8<z<0.8)=P(z<0.8)-P(z<-0.8)=0.7881-0.2119=0.5762

(b) If 100 people are randomly selected, find the probability that their mean IQ score is greater than 103. (Show work)

z=(103-100)/(15/sqrt(100))=2

P(z>2)=1-P(z<2)=1-0.9773=0.0227.

2.Imagine you are in a game show. There are 4 prizes hidden on a game board with 16 spaces. One prize is worth $4000, another is worth $1500, and two are worth $1000. You have to pay $50 to the host if your choice is not correct.

(a) What is your expected winning in this game? (Show work)

The chance of winning 4000 is 1/16, the chance of winning 1500 is 1/16 and the chance of winning 1000 is 1/8. The chance of losing: 1-1/16-1/16-1/8=3/4.

Therefore, the expected winning: 1/16*4000+1/16*1500+1/8*1000-3/4*50=431.35 dollars.

(b) If you are offered a sure prize of $400 in cash, and you can just take the money without playing the game. What would be ...

#### Solution Summary

The expected results and probability are found. The expected winning games are determined.