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Chapter 6 Learning Objectives
Compute probabilities using the probability
distribution of a discrete random variable.
Calculate and interpret the mean (expected
value) of a discrete random variable.
Calculate and interpret the standard deviation
of a discrete random variable.
Compute probabilities using the probability
distribution of certain continuous random
variables.
Describe the effects of transforming a random
variable by adding or subtracting a constant
and multiplying or dividing by a constant.
Find the mean and standard deviation of the
sum or difference of independent random
variables.
Find probabilities involving the sum or
difference of independent Normal random
variables.
Determine whether the conditions for using a
binomial random variable are met.
Compute and interpret probabilities involving
binomial distributions.
Calculate the mean and standard deviation of a
binomial random variable. Interpret these
values in context.
Find probabilities involving geometric random
variables.
*When appropriate, use the Normal
approximation to the binomial distribution to
calculate probabilities.
Section
Related
Example
on Page(s)
Relevant
Chapter
Review
Exercise(s)
6.1
349
R6.1
6.1
350, 352
R6.1, R6.3
6.1
353
R6.1, R6.3
6.1
355, 357
R6.4
6.2
365, 366, 368
R6.2, R6.3
6.2
372, 373, 374,
377
R6.3, R6.4
6.2
380, 381
R6.4
6.3
388
R6.5
6.3
390, 393, 396
R6.6
6.3
399
R6.5
6.3
406
R6.7
6.3
403
R6.8
Can I do
this?
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6.1
Discrete Random Variables
Read 346-350
What is a random variable? What is a probability distribution?
Give some examples of a random variable.
What is a discrete random variable?
Give some examples.
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How many times upside down? Imagine selecting a roller coaster in Missouri at random. Define the random
variable X = number of inversions (the times the roller coaster turns a rider upside down) on the randomly
selected ride. The table below gives the probability distribution of X, based on rcdb.com in 2015.
0
1
2
3
4
5
Inversions:
Probability: 14/22 1/22 0 3/22 2/22 2/22
(a) Show that the probability distribution for X is legitimate.
(b) Make a histogram of the probability distribution. Describe what you see.
(c) What is the probability that a randomly selected roller coaster has at least 3 inversions?
More than 3?
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“Over-Under” is played at many carnivals. It is a simple game played with 2 dice. Players place bets on one of 3
spaces, indicating that the player believes that the dice will land with a sum that is “over 7”, equal to 7, or
“under 7”. A winner receives his $1 back plus some additional amount ($4 for a 7 bet and $1 for the other 2
possible bets) and the carnival keeps the original bet for all other players. Suppose that a player places a $1 bet
on the “under 7” space. If X = net gain from a single $1 “under 7” bet, the possible outcomes are x = –1 or x = 1.
Here is the probability distribution of X:
Value:
–$1
$1
Probability: 21/36 15/36
If a player were to make this $1 bet on “under 7” many many times, what would the player’s average gain be?
Read 344-346
How do you calculate the mean (expected value) of a discrete random variable? Is the formula on the formula
sheet?
How do you interpret the mean (expected value) of a discrete random variable?
Calculate and interpret the mean of the random variable X in the roller coaster example on the previous page.
Does the expected value of a random variable have to equal one of the possible values of the random variable?
Should expected values be rounded?
HW #44 page 359 (1–13 odd, 31–34)
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6.1 continued
Read 352–354
How do you calculate the variance and standard deviation of a discrete random variable? Are these formulas on
the formula sheet?
How do you interpret the standard deviation of a discrete random variable?
The “equal to 7” and “under 7” bets in Over-Under both have the same expected value. How do you think their
standard deviations compare? Calculate them both to confirm your answer.
Value:
–$1
$4
Probability: 30/36 6/36
(=7)
(<7)
Value:
–$1
$1
Probability: 21/36 15/36
Use your calculator to calculate and interpret the standard deviation of X in the roller coaster example.
Are there any dangers to be aware of when using the calculator to find the mean and standard deviation of a
discrete random variable?
What is a continuous random variable? How do we find probabilities for continuous random variables?
Give some examples of continuous random variables.
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Is it possible to have a shoe size = 8? Is it possible to have a foot length = 8 inches?
How many possible foot lengths are there? How can we graph the distribution of foot length?
For each density curve below, find P(X>45).
For a continuous random variable X, how is P(X < a) related to P(X ≤ a)?
Heights of Two-Year-Old Males
The heights of two-year-old males closely follow a Normal distribution with a mean of  = 34 inches and a
standard deviation of  = 1.4 inches. Randomly choose one two-year-old male and call his height X.
(a) Find the probability that the randomly selected two-year-old male has a height of at least 33 inches.
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(b) Find the probability that a randomly selected two-year-old male’s height is between 32 and 36 inches.
(c) If P(X < k) = 0.75, find the value of k.
HW #45: page 359 (15, 17, 19, 21, 23, 25, 27–30)
6.2 Transforming Random Variables
The manager of a pizzeria noticed that, for the 30” colossal pizza, people add 0 (cheese pizza) to 6 toppings.
The number of toppings X beyond cheese that a randomly selected order has on a pizza has the following
distribution.
Number of Toppings (X)
1
2
3
4
5
6
Probability
0.20 0.30 0.20 0.15 0.10 0.05
Calculate and interpret the mean and standard deviation of X.
At that pizzeria, the charge for each topping (beyond cheese) on the 30” colossal pizza is $3 per topping. So, if
T = topping charge for a randomly selected pizza order, T = 2X. Here’s the probability distribution for T:
Topping Charge (T)
3
6
9
12
15
18
Probability
0.20 0.30 0.20 0.15 0.10 0.05
Calculate the mean and standard deviation of T.
What is the effect of multiplying or dividing a random variable by a constant?
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At this pizzeria, a 30” colossal cheese pizza costs $50. If C = overall cost for a randomly selected 30” colossal
pizza that has at least one topping beyond the basic cheese, C = 50 + 3T. Here is the probability distribution for
C:
Overall Cost (C)
Probability
53
56
59
62
65
68
0.20 0.30 0.20 0.15 0.10 0.05
Calculate the mean and standard deviation of C.
What is the effect of adding (or subtracting) a constant to a random variable?
What is a linear transformation? How does a linear transformation affect the mean and standard deviation of a
random variable?
For a certain car model, the distribution of X = hours to replace a recalled air conditioner part was
approximately normally distributed with a mean of 2.3 and a standard deviation of 0.4. The service manager at
a car dealership conducting the recall repair charges the manufacturer $60 per hour for labor and $140 for the
part.
(a) Define the variable Y to be the total amount charged to the manufacturer for a randomly selected air
conditioner recall repair for this model of car. Find the mean and standard deviation of Y.
(b) What is the probability that a randomly selected air conditioner recall repair for this car model has a total
repair charge of at least $280?
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HW #46 page 382 (35, 37, 39, 40, 41, 43, 45)
6.2 Combining Random Variables
Suppose that at a local soft-serve ice cream shop, the weight C of small cones follows a Normal distribution
with a mean of 3 ounces and a standard deviation of .3 ounces and the weight S of small sundaes follows a
Normal distribution with a mean of 5 ounces and a standard deviation of .4 ounces. What is the probability that
a randomly selected small sundae weighs less than a randomly selected small cone?
Simulation approach: (Fathom sundae-cone)
Based on the simulation, what conclusions can we make about the shape, center, and spread of the distribution
of a difference (and sum) of Normal random variables?
Non-simulation approach (don’t worry about the four steps):
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Suppose a breed of chicken lays eggs with weights that are Normally distributed with mean 2 ounces and
standard deviation 0.3 ounces. Using 3 randomly selected eggs of this type, what is the probability that a 3 egg
omelet (before any filling) will weigh less than 5 ounces?
Let D = the delivery charge for a randomly 30” colossal pizza from the pizzeria we examined earlier.
D  3.00 and  D  0.25 . Recall from earlier that C = overall cost for a randomly selected 30” colossal pizza
with X number of toppings and C = 58.40 and  C = 4.31. Find the mean and standard deviation of the cost of
a randomly selected 30” colossal pizza with delivery (C + D). What is the shape of the distribution?
HW #47: page 382 (47-57 odd, 58, 59, 61, 63, 65, 66)
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6.3 Binomial Distributions
Read 386-390
What are the conditions for a binomial setting?
What is a binomial random variable?
What are the parameters of a binomial distribution?
What are the possible values of a binomial random variable?
What is the most common mistake students make on binomial distribution questions?
Determine whether the random variables below have a binomial distribution. Justify your answer.
(a) Play Over-Under 12 times and let X = the number of times the dice display a sum equal to 7.
(b) Practice playing pool/billiards by taking 20 shots from various locations on the pool table. Let Y = number
of shots made.
(c) From a box of Froot Loop cereal, randomly select 80 loops and let C = color.
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Rolling Sixes in Yahtzee: The probability of rolling a six on a fair die is 1/6. If X = the number of sixes in a
roll of 5 fair dice, then X is binomial with n = 5 and p = 1/6.
What is P(X = 0)? That is, what is the probability that all 5 rolls are not sixes?
What is P(X = 1)?
What about P(X = 2), P(X = 3), P(X = 4), P(X = 5)?
In general, how can we calculate binomial probabilities? Is the formula on the formula sheet?
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Over-Under Revisited: In Over-Under, there is a 15/36 chance that the sum on the two dice is “over 7”. Suppose
you observe the next 10 rolls of the dice.
(a) What is the probability that exactly 4 of the rolls have a sum “over 7”?
(b) What is the probability that at least 8 of the rolls have a sum “over 7”?
HW #49: page 410 (69–79 odd)
6.3 More about the Binomial Distribution
How can you calculate binomial probabilities on the calculator?
Is it OK to use the binompdf and binomcdf commands on the AP exam?
How can you calculate the mean and SD of a binomial distribution? Are these on the formula sheet?
Over-Under Again: Let X = the number of the next 12 rolls of the dice in Over-Under that have a sum “over 7”.
(a) Calculate and interpret the mean and standard deviation of X.
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(b) How often will the number of rolls that sum “over 7” be within one standard deviation of the mean?
Read 401-404
When is it OK to use the binomial distribution when sampling without replacement?
Why is this an issue?
In some casinos, 8 decks of cards are shuffled together to deter players from trying to “count cards”. If a dealer
shuffles 8 decks together thoroughly and selects 2 cards at random, what is the probability of getting 2 hearts?
HW #50 page 410 (81–89 odd)
6.3 The Geometric Distribution
Read 404–405
What are the conditions for a geometric setting?
B- Binary
I- Independent trials
O- Open-ended number of trials
S- Same probability of success on each trial
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What is a geometric random variable? What are the possible values of a geometric random variable?
What are the parameters of a geometric distribution?
Trouble: In the board game Trouble, in order to move out of “Home” and into the “Start” space, one must press
the pop-o-matic dome and get the die to display a 6. The probability of rolling a 6 is 1/6.
(a) Explain why this is a geometric setting.
(b) Define the geometric random variable and state its distribution.
(c) Find the probability that it takes exactly three rolls to move out of “Home”.
(d) Find the probability that it takes at most three rolls to move out of “Home”.
In general, how can you calculate geometric probabilities? Is this formula on the formula sheet?
On average, how many rolls should it take to move out of “Home” in Trouble?
In general, how do you calculate the mean of a geometric distribution? Is the formula on the formula sheet?
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What is the probability it takes longer than average to move out of “Home”? What does this probability tell you
about the shape of the distribution?
HW #51: page 410 (93, 95, 97, 99, 101–105)
Review / FRAPPY!
FRAPPY: 2010 #4 (sampling car owners)
HW #52: page 416 Chapter Review Exercises (skip R6.8)
Review
HW #53: Page 418 Chapter 6 AP Practice Test
Chapter 6 Test
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