Math 141 Week-In-Review
8.4-8.6
1. Let the random variable X denote the number of boys in a six-child family. If the probability of a male birth is
.508,
a. Find the probability of exactly 3 boys in a six-child family.
n = 6 ; p = .508; r = 3; Answer from BINOM program: .3123
b. Find the probability of at least 4 boys in a six-child family.
n = 6 ; p = .508; lower r = 4; upper r = 6; Answer from BINOM program: .3589
c. Construct the binomial distribution and draw the histogram associated with the experiment.
d. Compute the mean and standard deviation of the random variable associated with this experiment.
Mean  3.048   1.2246
2. Daniel is throwing paper wads at a trash can. The probability the wad lands in the trash can is 0.675. If he
throws 20 wads at the trash can, what is the probability he gets at least 14 into the trash can?
N = 20; p = .675; lower r = 14; upper r = 20; Answer from BINOM program: .5114
3. If a fair die is rolled five times, what is the probability of rolling:
a. Exactly two 3’s?
n = 5 ; p = 1/6; r = 2; Answer from BINOM program: .1608
b. At least two 3’s?
n = 5 ; p = 1/6; lower r = 2; upper r = 5; Answer from BINOM program: .1962
4. The probability of recovering after a particular type of operation is .5. What is the probability that at least 5
of 10 patients undergoing the operation will recover?
n = 10; p = .5; lower r = 5; upper r = 10; Answer from BINOM program: .6230
5. A manufacturing process produces, on the average, 3% defective items. The company ships 10 items in
each box and wishes to guarantee no more than 1 defective item per box. If this guarantee accompanies
each box, what is the probability that the box will fail to satisfy the guarantee?
n = 10; p = .03; lower r = 2; upper r = 10; Answer from BINOM program: .0345
6. A person with tuberculosis is given a chest x ray. Four tuberculosis x-ray specialists examine each x ray
independently. If each specialist can detect tuberculosis 80% of the time when it is present, what is the
probability that at least 1 of the specialists will detect tuberculosis in this person?
n = 4; p = .8; lower r = 1; upper r = 4; Answer from BINOM program: .9984
7. A manufacturing process produces light bulbs with life expectancies that are normally distributed with a
mean of 500 hours and a standard deviation of 100 hours. What percentage of the light bulbs can be
expected to last between 500 and 670 hours?
normalcdf (500, 670, 500, 100)  .4554 or 45.54%
8. Use the normal distribution to approximate the following: A credit card company claims that their card is
used by 40% of the people buying gasoline in a particular city. A random sample of 20 gasoline purchasers
is made. If the company’s claim is correct, what is the probability that:
a. From 6 to 12 people in the sample use the card?
 = 20(.4) = 8; x = 20(.4)(.6)  2.1090; normalcdf (5.5, 12.5, 8, 2.1909)  .8531
b. Fewer than 4 people in the sample use the card?
Normalcdf (-.5, 3.5, 8, 2.1909)  .0199
9. Use the normal distribution to approximate this binomial distribution: A company manufactures 50,000
ballpoint pens each day. The manufacturing process produces 50 defective pens per 1,000, on the
average. A random sample of 400 pens is selected from each day’s production and tested. What is the
probability that the sample contains
a. At least 14 and no more than 25 defective pens?
 = 400(.05) = 20; x = 400(.05)(.95)  4.3589; normalcdf (13.5, 25.5, 20, 4.3589)  .8285
b. 33 or more defective pens?
normalcdf (32.5, 400.5, 20, 4.3589)  .0021
10. The average healing time of a certain type of incision is 240 hours, with standard deviation of 20 hours.
What percentage of the people having this incision would heal in 8 days or less? Assume a normal
distribution.
normalcdf (0,192, 240, 20)  .0082 or .82%
11. The average height of a hay crop is 38 inches, with a standard deviation of 1.5 inches. What percentage of
the crop will be 40 inches or more? Assume a normal distribution.
normalcdf (40, 1E99, 38, 1.5)  .0912
12. Let Z be the standard normal variable. Find z if z satisfies the value:
a. P(Z < z) = .8532
b. P(Z > z) = .8532
invNorm (.8532)  1.05
invNorm (1 - .8532)  -1.05
c. P(-z < Z < z) = .8532 - invNorm (.5 - .8532/2)  1.45