Math 243
Chapter 16
Probability Models
Overview
 Bernoulli Trials
 The Geometric Model
 The Binomial Model
 Normal Approximations to the Binomial Model
Example 1. Roll a standard die 3 times. Let the random variable 𝑋 represent the number of times a 5 is
rolled. (So, we will call a success is when a 5 is rolled in a single trial.)
a. What is the probability of success in each trial?
b. What is the probability of failure in each trial?
c. What is the probability of rolling the first five on the third roll?
Bernoulli Trials
Repeated trials of an experiment are called Bernoulli trials if the following conditions are met:
1. Each trial has only two possible outcomes (generally designated as success and failure)
2. The probability of success, 𝑝, remains the same for each trial
3. *The trials are independent. (The outcome of one trial has no influence on the next)
*The 10% condition: Bernoulli trials must be independent. When selecting items without replacement,
we know they are not independent. However, if we are selecting less than 10% of the population it is
okay to assume independence and proceed with this model.
Example 2. Determine which of the following situations involve Bernoulli trials.
a. You are rolling 5 dice and need to get at least two 6’s to win the game.
b. We record the distribution of eye colors found in a group of 500 students.
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c. A manufacturer recalls a doll because about 3% have buttons that are not properly attached.
Customers return 37 of these dolls to the local toy store. Is the manufacturer likely to find any
dangerous buttons?
d. A city council of 11 Republicans and 8 Democrats picks a committee of 4 at random. What’s the
probability that they choose all democrats?
Geometric Probability Model for Bernoulli Trials
𝑿~𝑮𝒆𝒐𝒎(𝒑)
𝑝 = probability of success
𝑞 = (1 − 𝑝) = probability of failure
𝑋 = the number of trials until the first success
Probability of first success on the xth trial:
𝑃(𝑋 = 𝑥) = 𝑞 𝑥−1 𝑝
Expected Value:
𝐸(𝑋) = 𝜇 = 𝑝
Standard Deviation:
𝜎 = √𝑝2
1
𝑞
Example 3. A basketball player makes about 82% of her free throws. Assuming the shots are
independent, find the probability that in tonight’s game she will do the following:
a. Make the first free throw on the 4th attempt?
b. Make the first free throw on the 12th attempt?
c. What is the expected number of shots it will take until she makes a free throw?
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In Example 3, we looked at the probability that the basketball player made her first basket on the 4th
try. What if we wanted to know the probability that she made exactly one basket in the first four
attempts? (Note: Which of the first four doesn’t matter, just that she makes one!) Write down all
possible combinations that she makes one basket out of four:
Combinations
Each different order in which we can have k successes in n trials is called a combination. We denote
𝑛
the total number of ways this can happen as (𝑘 ) or as 𝑛𝐶𝑘 , where 𝑛𝐶𝑘
=
𝑛!
𝑘!(𝑛−𝑘)!
.
Example 4. Calculate the following by hand.
a. 4C1
b. 7C2
Example 5. Check each of the above using your calculator.
TI-84+ Type the first number, n. Then press 2nd CATALOG and use the LOG key to skip to letter N.
Select nCr, type the second number, k, and press enter.
a. 4 nCr 1 =
b. 7 nCr 2 =
TI-89 Titanium Press CATALOG and type 6 to skip to the letter N. Select nCr(, type 4,1) and press
enter.
a. nCr(4,1) =
b. nCr(7,2) =
c. 20C3 =
d. 100C5 =
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Binomial Probability Model for Bernoulli Trials
𝑿~𝑩𝒊𝒏𝒐𝒎(𝒏, 𝒑)
𝑝 = probability of success
𝑞 = (1 − 𝑝) = probability of failure
𝑋 = the number of successes in n trials
Probability of x successes in n trials:
𝑃(𝑋 = 𝑥) =
𝑛𝐶𝑥
𝑛𝐶𝑥 𝑝
𝑥 𝑛−𝑥
𝑞
𝑛!
= (𝑛𝑥) = 𝑥!(𝑛−𝑥)!
Expected Value:
𝐸(𝑋) = 𝜇 = 𝑛𝑝
Standard Deviation:
𝜎 = √𝑛𝑝𝑞
Example 3 Continued. Now with the Binomial model we can calculate the probability that our
basketball player will make exactly 1 shot in 4 free throws.
Example 6. Our freethrower is still making 82% of her baskets. Assume each shot is independent of
the last. She’s going to shoot 10 free throws.
a. What’s the probability that she makes exactly 5 free throws?
b. What’s the probability that she makes 9 or 10 free throws?
c. What’s the probability that she makes 2 or fewer free throws?
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d. What’s the expected number of baskets she makes? What’s the standard deviation?
Finding Binomial Probabilities on the Calculator
BinomalPDF(n, p, x) give the probability of x successes in n trials
TI-84+ BinomialCDF(n, p, x) gives the probability of 0 to x successes.
TI-89 Titanium BinomialCDF(n, p, lower value, upper value) gives the probability of lower to
upper successes.
Check the answers for Example 6 using your calculator:
a.
b.
c.
Example 7. Binomial, geometric or neither?
a. A worker opening oysters to look for pearls counts the number of oysters they have to open until
they find the first pearl.
b. A quality control inspector takes a random sample of 20 items from a large lot, inspects each item,
classifies each as defective or non-defective, and counts the number of defective items in the sample.
c. An engineer chooses and SRS of 10 switches from a shipment of 10,000 switches. Suppose 10% of
the switches in the shipment are bad. The engineer counts the number of bad switches in the sample.
d. A supervisor at the end of an assembly line counts the number of non-defective items produced
until she finds a defective one.
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Normal Approximations for Binomial Probability Models
So far we’ve use the binomial probability model to determine, for example, the probability that a
basketball player makes 7 out of 10 baskets. We also determined the odds that she made 9 or 10
baskets out of 10, and 2 or fewer baskets out of 10.
If we want to determine the probability that she made at least 125 baskets out of 200, we could
technically do this using the Binomial probability model. Keeping our assumption that she makes 68%
of the baskets that she shoots and that each shot is independent of the last, here’s what it would look
like:
𝑃(𝑋 ≥ 125) = 𝑃(𝑋 = 125) + 𝑃(𝑋 = 126) + 𝑃(𝑋 = 127) + · · · + 𝑃(𝑋 = 200)
= 200C125(0.82)125(0.18)75 + 200C126(0.82)126(0.18)74 + … + 200C200(0.82)200(0.18)75
Just try calculating 200C125. What happens?
Success/Failure Condition
A Binomial model is approximately Normal if we expect at least 10 successes and 10 failures,
determined by:
𝑛𝑝 ≥ 10 and 𝑛𝑞 ≥ 10
Note: We MUST show that we’ve verified this condition every time we approximate a Binomial model
with a Normal model.
𝑩𝒊𝒏𝒐𝒎(𝒏, 𝒑) ~ 𝑵𝒐𝒓𝒎𝒂𝒍(
,
)
Example 8. A basketball player makes about 82% of her free throws. Assume each shot is independent
of the last. She shoots 200 free throws.
a. Write down the binomial model for this scenario. Then write down the Normal approximation
model for this scenario.
First check the success/failure condition:
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b. What is the probability that she will make at least 125 of the 200 free throws?
c. Can we use a Normal approximation to calculate the likelihood that this basketball player makes
exactly 125 out of 200 baskets?
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Example 9. A coin is flipped 1000 times. What’s the probability that fewer than 450 heads are flipped?
Check the success/failure condition:
Example 10. A company reports that their computer chips fail 2% of the time. An independent
contractor tests a batch of 3000 chips and finds that 94 of them fail. Does the company’s reported
failure rate of 2% seem reasonable?
Check the success/failure condition:
To Sum it All up: PROBABILITY MODELS



Geometric (how many trials until the 1st success)
Binomial (how many successes in n trials)
Normal Model (to approximate a Binomial model when we expect at least 10 successes
and 10 failures)
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More Practice
1. An Olympic Archer is able to hit the bull’s-eye 80% of the time. Assume each shot is independent of
the others. If he shoots 6 arrows, what’s the probability of each of the following results?
a. His first bull’s-eye comes on the third arrow.
b. He misses the bull’s-eye at least once.
c. His first bull’s-eye comes on the fourth or fifth arrow.
d. He gets exactly 4 bull’s-eyes
e. He gets at least 4 bull’s-eyes.
f. He gets at most 4 bull’s-eyes
g. How many bull’s-eyes do you expect him to get?
h. With what standard deviation?
i. If he keeps shooting arrows until she hits the bull’s-eye, how long do you expect it will take?
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2. Suppose a computer chip manufacturer rejects 3% of the chips produced because they fail presale
testing. What is the probability that the seventh chip you test is the first bad one you find? How many
chips would you expect it to take to find a bad one?
3. Ken Griffey Jr. has a lifetime batting average of .305. (This is the probability of getting a hit). If he
batted 5 times in one game, what is the probability that he gets at least 3 hits?
4. Suppose 19% of students at one college have high blood pressure. If you keep picking students at
random from this college, how many students do you expect to test before finding one with high
blood pressure?
5. A 2002 Rutgers University study found that 74% of high school students have cheated on a test at
least once. A local high school principal conducts a surveys of 481 students. What’s the probability
that 350 or more students have cheated? (What tells you to use a Normal approximation here?)
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