Discrete Random Variables & The Binomial Dist’n
Name _______________________________
Chapter 6 Homework
Read each problem carefully. Write your answer in the blank, or circle the correct answer.
1. What are the requirements for a discrete probability distribution?
(circle all that apply)
a)
b)
c)
d)
The sum of the probabilities = 0.
0 ≤ P(x) ≤ 1
The sum of the probabilities = 1.
0 < P(x) < 1
2. Determine if the following random variable is discrete or continuous. In each case, state the possible
values of the random variable.
The number of fish caught during a fishing tournament.
a)
b)
c)
d)
Discrete; Possible values are 0, 1, 2, 3, …
Continuous; Possible values are 0, 1, 2, 3, …
Discrete; Possible values are x ≥ 0.
Continuous; Possible values are x ≥ 0.
The amount of snowfall in inches.
a)
b)
c)
d)
Discrete; Possible values are 0, 1, 2, 3, …
Continuous; Possible values are 0, 1, 2, 3, …
Discrete; Possible values are x ≥ 0.
Continuous; Possible values are x ≥ 0.
3. Determine the missing value in the following discrete probability distribution.
x
3
4
5
6
P(x)
0.17
0.53
0.07
4. Suppose a baseball player had 236 hits in a season. In the given probability distribution, the random
variable x represents the number of hits the player obtained in a game.
x
P(x)
0
0.1293
1
0.4377
2
0.2119
3
0.1787
4
0.0136
5
0.0288
Compute the mean and standard deviation of the number of hits.
Mean = ____________ (1 decimal)
Standard Deviation = ______________ (1 decimal)
5. Suppose a life insurance company sells a $220,000 one-year term life insurance policy to a 23-year
old female for $360. The probability that the female survives the year is 0.999562. Fill out the
following probability distribution then compute the expected value of this policy to the insurance
company (ie, the profit).
Outcome
Profit ($)
Probability
Expected Value = $_______________
(round to the nearest cent)
Which of the following interpretations is correct?
a) The insurance company expects to make an average profit of $263.64 on every 23-year-old
female it insures for 1 year.
b) The insurance company expects to make an average profit of $32.71 on every 23-year-old
female it insures for 1 month.
c) The insurance company expects to make an average profit of $23.97 on every 23-year-old
female it insures for 1 month.
d) The insurance company expects to make an average profit of $359.84 on every 23-year-old
female it insures for 1 year.
6. In the game of roulette, a player can place an $8 bet on the number 17 and have a 1/38 probability of
winning. If the metal ball lands on 17, the player is awarded $280. Otherwise, the player gets
nothing and the casino keeps the $8. Fill out the following probability distribution then compute the
expected value of the game to the player (ie, the expected winnings).
Outcome
Winnings ($)
Probability
(fraction, do not round!)
Expected Value = $__________ __
(round to the nearest cent)
If you played the game 100 times, how much would you expect to lose?
________________
7. Determine if the following probability experiments represent binomial experiments.
A random sample of 15 college professors is obtained, and the individuals selected are asked to
state their heights.
a) No, this is not binomial because the variable is continuous and there are not two outcomes
for each trial.
b) Yes, this is binomial because each trial has two outcomes.
c) Yes, this is binomial because the probability of success is the same for each trial of the
experiment.
d) No, this is not binomial because the trials are not independent.
Five cards are selected from a standard 52-card deck with replacement. The number of kings
selected is recorded.
a)
b)
c)
d)
No, this is not binomial because the experiment is not performed a fixed number of times.
Yes, this is binomial because all criteria for a binomial experiment have been satisfied.
No, this is not binomial because there are more than two outcomes for each trial.
No, this is not binomial because the trials are not independent and the probability of a
success differs from trial to trial.
8. According to a report, 65.1% of murders are committed with a firearm.
If 400 murders are randomly selected, how many would we expect to be committed with a firearm?
We would expect _________________ .
(round to 1 decimal)
Would it be unusual to observe 289 murders by firearm in a random sample of 400 murders? Why?
(hint, you will need to calculate the standard deviation and consider the “usual” cutoffs)
a)
b)
c)
d)
e)
No, because 289 is greater than µ + 2σ
Yes, because 289 is between µ – 2σ and µ + 2σ.
Yes, because 289 is greater than µ + 2σ.
Yes, because 289 is less than µ – 2σ.
No, because 289 is between µ – 2σ and µ + 2σ.
9. A study showed that 67% of all senior citizens wear sunglasses while driving. Suppose 18 senior
citizens are selected at random and the number wearing sunglasses is recorded.
Calculate the probability that exactly 9 are wearing sunglasses.
a)
b)
c)
d)
.1019
.0614
.8981
.0272
Calculate the probability that at least 9 are wearing sunglasses.
a)
b)
c)
d)
.9728
.8981
.9595
.0405
Calculate the probability that fewer than 8 are wearing sunglasses.
a)
b)
c)
d)
.0133
.0405
.9594
.0272
For 500 randomly selected senior citizens, compute the mean and standard deviation of the number of
who wear sunglasses while driving.
Mean = ________________
Standard Deviation = _______________ (1 decimal)
Would it be unusual if 350 of the 500 senior citizens were wearing sunglasses while driving?
a)
b)
c)
d)
Yes, because 350 is more than 2 standard deviations from the mean.
Yes, because 350 is within 2 standard deviations of the mean.
No, because 350 is more than 2 standard deviations from the mean.
No, because 350 is within 2 standard deviations of the mean.