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AP Statistics
Notes 6.4
Binomial Setting – 4 conditions MUST be satisfied:
1. Binary – each observation falls into one of two categories: “success” or “failure”
2. Independent – knowing the result of one trial does not have any effect on the result of
any
other trial
3. Number – must be fixed in advance
4. Success – probability , p, of success is the same for each trial
Binomial Distribution: probability distribution of X with parameters n and p abbreviation: B(n, p)
Is this a binomial setting?
1. A balanced die is tossed 5 times; X is the number of times a 6 is rolled.
2. An SRS of 100 disposable razors; X is the number of defective razors. On average, 500 out of every
100,000 razors are defective.
3. An engineer chooses an SRS of 10 switches from a shipment of 10,000 switches. Suppose that
(unknown to the engineer) 10% of the switches in the shipment are bad. The engineer counts the
number X of bad switches in the sample.
4. Blood type is inherited. If both parents carry genes for the O and A blood types, then each child has
probability 0.25 of getting two O genes and so of having blood type O. Different children inherit
independently of each other. The number of O blood types among 5 children of these parents is the
count X of successes.
5. Deal 10 cards from a shuffled deck and count the number X of red cards.
Binomial Probability Distribution Function
OR
Binompdf
Examples:
1. Fifty-seven percent of companies in the U.S. use networking to recruit workers. What is the
probability that in a survey of ten companies exactly half of them use networking to recruit
workers?
2. If X represents the number of girls in families having four children, then X is a binomial
random variable with n = 4 and p = .5. What is the probability that a family has 2 girls?
Binomial Cumulative Distribution Function
OR
Binomcdf
for every possible value of k
3. Sixty percent of teenagers who drink alcohol do so because of peer pressure. In a sample of 15
teenagers who drink, find
a. P(X = 5)
b. probability that five or fewer do so because of peer pressure.
c. probability that five or more do so because of peer pressure.
d. probability that more than five do so because of peer pressure.
4. Approximately 10% of students are left-handed. If we select a random sample of 15 students,
find:
a. the probability that exactly 3 students in the sample are left-handed.
b. the probability that more than 3 students in the sample are left-handed.
c. the probability that at most 3 students in the sample are left-handed.
d. the probability that at least 3 students in the sample are left-handed.
Mean and standard deviation of a Binomial Random Variable:
 B  np
 B  np (1  p )
Examples:
5. If X represents the number of girls in families having four children, then X is a
binomial random variable with n = 4 and p = .5. What is the mean and standard
deviation?
Normal Approximation to Binomial Distributions
When n increases, the formula becomes awkward to use. That leaves us with two
options:
1. use software or statistical calculator
OR
2. as the number of trials n gets larger, the binomial distribution gets close to a
normal distribution
As n gets larger B(n,p)  N B ,  B 
***As a rule of thumb, we only use the normal approximation when n
and p satisfy np  10 AND n(1  p)  10
Examples.
6. Sixty percent of teenagers who drink alcohol do so because of peer pressure. Find
the normal approximation probability that in a sample of 25 teenagers who drink,
a. 20 or fewer do so because of peer pressure.
b. at least 20 do so because of peer pressure.