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Math 166, Fall 2011, Benjamin
Aurispa
2.4 Bernoulli Trials/Binomial Experiments
Experiments with exactly two outcomes are called Bernoulli trials or binomial trials. A sequence of
these binomial trials is called a binomial experiment. Its important to recognize when an experiment is
binomial because there is a quick way to calculate binomial probabilities.
Properties of Binomial Experiments:
1. The number of trials in the experiment is fixed.
2. There are two outcomes in the experiment: “success” and “failure.” (Note: Defining “success” and
“failure” varies according to each problem and what we are observing.)
3. The probability of success in each trial is the same.
4. The trials are independent of each other.
For example, if I toss a coin, a “success” could be getting a head, and then a “failure” would be getting a
tail. The probability of success in this example would be 12 .
Another example: If an experiment consists of rolling a die and observing if a 6 is rolled, I could say that
I have a “success” if I roll a 6, in which case a “failure” would be rolling anything other than a 6. So, the
probability of a success in this example would be 16 .
Examples: Determine whether the following experiments are binomial experiments. If not, why not?
• Rolling a die until a 5 is rolled.
• Rolling a die 3 times and observing the number rolled each time.
• Selecting 7 cards one at a time with replacement out of a standard deck and observing if each is black
or red.
• Selecting 4 cards one at a time without replacement and observing if each card is black or red.
• Pulling 5 marbles one at a time with replacement from a jar consisting of 6 red, 5 blue, and 4 yellow
marbles, and observing if the marble is red or not each time.
In a binomial experiment, we are interested in the probability that a certain number of successes occurs. We
let X represent the number of successes so that P (X = r) represents the probability of having r successes.
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Math 166, Fall 2011, Benjamin
Aurispa
To find a binomial probability, we only need to know 3 things:
n = total number of trials,
p = probability of success,
r = number of desired successes
Then, the probability of getting exactly r successes in n trials, P (X = r), is:
P (X = r) = C(n, r)pr q n−r
where n, p, and r are as defined above, and q is the probability of failure. Since the probability of success is
p, the probability of failure, q, is 1 − p.
To find the probability of exactly r successes in a binomial experiment, we can use the binompdf command
on the calculator instead of the formula.
1. Press 2nd VARS (which is DISTR).
2. Choose binompdf.
3. Then, the probability of exactly r successes in n trials is:
P (X = r) = binompdf (n, p, r)
Example: The probability a basketball player makes a free throw is 0.72. If he shoots 12 free throws in a
game, what is the probability he will make exactly 7 of them?
What if we want to find the probability that he makes at most 7 free throws?
We could add up P (X = 0) + P (X = 1) + · · · + P (X = 7), but there is another command on the calculator
that will do this for us: binomcdf. This command is found on the same menu as binompdf and is option
binomcdf.
In general, the probability of at most r success in n trials is:
P (X ≤ r) = binomcdf (n, p, r)
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Math 166, Fall 2011, Benjamin
Aurispa
What if I want to find the probability that he makes at least 7 free throws, P (X ≥ 7)? There is no specific
command on the calculator for this, but we can use the idea of complements to solve using the binomcdf
command.
More Examples:
1. A fair die is cast 9 times. What is the probability of obtaining
(a) Exactly 2 6’s?
(b) More than three 6’s?
2. A quiz consists of 10 multiple choice questions. Each question has five possible answers. In order to
pass the quiz, a student must get at least 6 correct. If a student guesses at each question, what is the
probability that they will pass the quiz?
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Math 166, Fall 2011, Benjamin
Aurispa
3. Suppose that 30% of the restaurants in a certain part of a town are open past 10pm. If 5 restaurants
are randomly selected, what is the probability that
(a) None or all of the restaurants are open past 10pm?
(b) No more than 3 of the restaurants are open past 10pm?
4. The manager of a movie rental store knows that 40% of the people who are browsing in the store will
actually rent a movie. What is the probability that among 20 people who are browsing in the store...
(a) Fewer than five will rent a movie?
(b) More than 3 but fewer than 10 will rent a movie?
5. The probability that a DVD player produced by a company is defective is estimated to be 0.09. If
a sample of 14 DVD players is selected at random, what is the probability that the sample contains
between 2 and 10 defectives inclusive, that is, at least two but no more than 10.
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