AP Statistics
Probability Test #2 Review – Simulations, Expected Values, Variance, Standard Deviation
1. A study by Liu and Schutz (1994) states that in ice hockey, the better team has a 65% chance of scoring the first goal in overtime of
any game. If a team has five overtime games, and is considered the better team in all of them, what is the probability that this team
will score the first overtime goal in at least four out of these 5 games? Answer the question using 15 trials of a simulation based on the
random number table below. Make your procedure clear by following all the required steps.
84177 06757 17613 15582 51506 81435 41050 92031 06449 05059 59884
57967 05811 84514 75011 13006 63395
31180 53115 84469
55041 15866 06589 13119 71020 85940 91932
54355 52704 90359 02649 47496 71567 94268 08844 26294 64751 08989
09488
57024 74987 94868
2. Charlie Sheenenjoys eating M&M’s like a winner in his winning fashion where he grabs an M&M at random from the bowl and
then proceeds to eat all of the remaining M&M’s in that particular color. Then he repeats the process until all six colors of M&M’s
area eaten. If the proportion of green M&M’s in any given bag is 14%, what is the probability that the greens will be eaten first or
second? Run a simulation of 25 trials using the random number table provided below.
54401 95654 71896 90054 35580 25562 82173 85782 48397 87109 97531 13149 92854
53738 71087 72155 68951 78564 23166 36243 50129 08472 33443 31971 62947 70019
75360 59338 07404 36648 22357 91786 56942 61474 00995 55874 64678 91382 84714
51602 77496 26371 26630 78968 95395 19558 73628 11417 71492 17826 52265 02063
52670 74291 62542 97790 65487 09281 77900 98599 09944 02467 24898 57606 86041
3.
A young woman works two jobs and receives tips from both jobs. As a hairdresser, her distribution of weekly tips has a mean of
$65 and a standard deviation of $5.75. As a waitress, her distribution of weekly tips has a mean of $164 with a standard deviation
of $8.02. What are the mean and standard deviation of her combined weekly tips? Assume that the tips received for each job are
independent. Lastly, find the probability that she will earn at least $250 in any given week?
4.
In a classroom activity, students place one hand into a container full of pennies and try to grab as many as they can. The number
of coins grabbed by boys has a normal distribution with a mean of 32 coins and a standard deviation of 6 coins. The number of
coins grabbed by girls is also normally distributed with a mean of 23 and a standard deviation of 4 coins. One randomly selected
boy and one randomly selected girl each grab a handful of coins. If the number of coins drawn by each is independent, what is the
approximate probability that the boy grabs no more than 5 coins more than the girl?
5.
A television game show has three payoffs with the following probabilities:
Payoff (X):
$0
$1000
$10,000
a) What are the mean and standard deviation for the payoff variable X?
Probability:
.6
.3
.1
b) Define a new random variable Y = 5 – 2X based on the random variable X. What are the mean and standard deviation of the
random variable Y?
6.
A hit man has been hired to take out numerous individuals from an enemy mob during their monthly assembly. He will be paid
$5,000 for each foot soldier, $10,000 for the Consigliore, $20,000 for the Underboss, and $50,000 for the Boss. This particular
Mafia has 30 members total. The hit man has recently designed a new grenade type weapon that will fire 3 rounds simultaneously
in different directions, but each round is a mini-missile that will target and destroy the first individual it zeroes in on. The hit man
decides the best way to use the weapon is to drop it into the building from the roof in the middle of the assembly. Assuming that
the mobsters’ are seated in locations independent of each other, find the amount of money the hit man can expect to earn as well
as the standard deviation of his earnings.
7.
Shelby collects rare antique records to play on her Victrola. Rare 78s average $75 with a standard deviation of $25. While
browsing an antique shop she finds 6 records worthy of purchasing.
a) Find the expected amount and standard deviation of Shelby’s
purchase if the records are on sale for 60% off their regular price.
b) Find the expected amount and standard deviation of Shelby’s
purchase if the owner instead decides to mark down each record
by $8.
Helpful Formulas to remember:
Expected Values
( )
(
(
)
( )
( )
)
(
(
(
)
(
The expected value is the total sum of each x multiplied by the probability that it will occur.
∑ ( )
)
)
( )
)
The expected value of the sum of two random variables is the same as the sum of their respective
expected values.
Adding a constant to the random variable X simply adds the constant to the mean
( )
Multiplying the random variable X will also multiply the mean by the same constant.
Variance and Standard Deviation
( )
∑ ( ) (
)
The variance is the total sum of each x’s squared deviation from the mean multiplied by
the probability of that x occurring.
To get the Standard Deviation, take the square root of the Variance.
(
(
)
)
( )
( )
Multiplying the random variable X by a constant will scale the SD by the same constant,
but this will scale the Variance by the square of the constant.
NEVER add standard deviations together because they are square rooted values.
(
)
( )
( )
(
)
( )
( )
√
√
√
When adding OR subtracting two or more random variables, ALWAYS ADD their
variances (because we are less sure of the outcome – the variability increases). To get
the SD of a sum or difference, take the square root of this total.
Helpful Formulas to remember:
Expected Values
( )
(
(
(
)
( )
( )
)
(
(
)
(
The expected value is the total sum of each x multiplied by the probability that it will occur.
∑ ( )
)
)
( )
)
The expected value of the sum of two random variables is the same as the sum of their respective
expected values.
Adding a constant to the random variable X simply adds the constant to the mean
( )
Multiplying the random variable X will also multiply the mean by the same constant.
Variance and Standard Deviation
( )
∑ ( ) (
)
The variance is the total sum of each x’s squared deviation from the mean multiplied by
the probability of that x occurring.
To get the Standard Deviation, take the square root of the Variance.
(
(
)
)
( )
( )
Multiplying the random variable X by a constant will scale the SD by the same constant,
but this will scale the Variance by the square of the constant.
NEVER add standard deviations together because they are square rooted values.
(
)
( )
( )
(
)
( )
( )
√
√
√
When adding OR subtracting two or more random variables, ALWAYS ADD their
variances (because we are less sure of the outcome – the variability increases). To get
the SD of a sum or difference, take the square root of this total.