# Binomial & Normal Probabilities

1- Given the following set of numbers assume this is a population: 4,5,7,7,3,5,8,9,6.

If this set of numbers were a sample, what divisor would you use to find the variation?

2- Earlier this semester we had a problem with birds with sore throats. I own the pet store that has these birds and I need to get them some medical treatment. When I look in the phone book, I find four veterinary hospitals/clinics with doctors that treat birds. They are: a) Charlotte Parker's Animal Hospital (C P A H)

b) Maggie Nash's Veterinary Clinic (M N V C)

c) Mark Galluzzo's Home for Wobbly Wings (M G H W2) and

d) Rachana Dangol's Tiny Animal Shelter (R A T S)

The phone book does not list a phone number for any of these organizations. However, it does list 50 separate phone numbers for "doctors" who treat birds, but only lists the single title "bird-doctor" and a phone number beside each number. It does not indicate any more information than that. These 50 doctors work in the four organizations listed above, but I cannot tell which organization each belongs to, because all the phone book has is a number. Some of the doctors have vaccines that heal the sore throats; some of them do not. Additionally, one of the doctors, one Elizabeth Guerrier, has the vaccine but is afraid of birds and won't treat them. The distribution of doctors in the four hospital/clinics is as follows: C P A H M N V C M G H W2 R A T S

Doctors with vaccine 0 4 6 5

With vaccine but afraid 1

Doctors without vaccine 8 8 6 12

If I randomly dial one of the doctor's numbers, what is the probability that:

a) I get a doctor working with Charlotte?

b) I get a doctor working with Charlotte or Maggie?

c) I get a doctor who does not have the vaccine?

d) I get a doctor working with Mark or one who does have the vaccine?

e) I get a doctor who, for whatever the reason, is unable to give the vaccine?

f) I get a doctor who has the vaccine?

If I dial two doctors' numbers in succession, making sure that I do not accidentally redial the first, what is the probability that:

a) I will get two doctors that do not have the vaccine?

b) I will not get a doctor with the vaccine on the first call and do get a doctor with the vaccine on the second call?

Who does Elizabeth Guerrier work for?

3- According to a study by the National Endowment for the Arts, 20% of U.S women attended a musical play in 2002.

a) What kind of a distribution is this?

b) In a random sample of 15 U.S women, what is the probability that exactly 5 attended a play in 2002?

c) In a similar random sample of 15 U.S women, what is the probability that fewer than 7 attended a play in 2002?

d) How would you find the probability that 7 or more women attended a play in 2002? (Just explain briefly; you do not have to find the probability.)

e) Find the mean and the standard deviation for this sample of 15 women.

4- Erjona Kerthi and her partner, Chris McNair, purchased a foreclosed property for $50,000 and spent an additional $27,000 on repairs. They feel that they have a 15% probability of reselling the property for $120,000, a 45% probability of reselling it for $100,000, a 25% probability of reselling it for $80,000, and a 15% probability of taking a loss and selling it for $60,000. What is their expected profit/loss for reselling the property?

5- In roulette a player can place a $5 bet on the number 17 and have a 1/38 probability of winning. If the player wins, he/she will receive $175.00. Otherwise, the house will take the $5.00. What is the expected value of the game to the player?

6-According to the police, 67% of all murders are committed with a firearm. A sample of 25 murders was selected, and the number of murders involving firearms was recorded. (Hint: The binomial table does not have an "n equal to 25, but are the products of "n" times "p" times "q" both greater than 5? How can you approach this problem a normal distribution approximation?)

a) Find the probability that exactly 22 murders were committed with a firearm.

b) Find the probability that between 14 and 16 murders (14 to 16 inclusive) were committed with firearms.

7- Steel rods are manufactured with a mean length of 25 cm. Because of the machinery tolerances, the lengths of these rods are approximately normally distributed with a standard deviation of .07 cm.

a) What is the probability that a randomly selected rod has a length less than 24.9 cm?

b) Any rods which are shorter than 24.85 cm or longer than 25.15 cm will be discarded. Approximately what percentage of the rods will be discarded?

8) The mean incubation time of fertilized chicken eggs maintained at 100.5 degrees F in a still-air incubator is 21 days. These incubation times are normally distributed with a standard deviation of 1 day.

a) What is the probability that a randomly selected fertilized egg hatch in less than 20 days?

b) What is the probability that a randomly selected egg takes over 22 days to hatch?

c) What is the probability that a randomly selected egg hatches between 19 and 21 days?

9- In a recent study of medical students, it was found that medical residents' average number of hours worked per week was 81.7. The number of hours worked per week by medical residents is approximately normally distributed with a standard deviation of 6.9 hours.

a) What is the probability that a randomly selected resident works more than 80 hours per week?

b) What is the probability that a randomly selected resident works more than 100 hours per week?

c) If a sample of 16 medical students is selected, what is the probability that the mean number of hours worked by the members of that sample is more than 80 hours per week?

#### Solution Summary

The solution provides step by step method for the calculation of binomial and normal probabilities. Formula for the calculation and Interpretations of the results are also included.