Chapter 2: Counting and Probability
Section 2.4
Bernoulli Trials
An experiment with two outcomes is called a
.
Example 1 The following are examples of Bernoulli trials:
• Flipping a coin and noting whether it lands heads up.
• Testing a person for a disease and noting whether it is positive.
• Picking a card out of a standard 52-card deck and noting whether the card is a face
card.
• Rolling a die and noting whether the uppermost face is a 6.
• Picking a marble out of a bag and noting whether the marble is purple.
We will mostly be concerned with Bernoulli trials that are repeated more than one time.
experiments (or Bernoulli experi-
These are called
ments or repeated Bernoulli trials).
Fundamental Assumption for Bernoulli Trials: Successive Bernoulli trials are independent of one another.
A binomial experiment has the following properties:
1.) The number of trials in the experiment is fixed.
2.) There are two possible outcomes of each trial:
and
.
3.) The probability of success in each trial is the same.
4.) The trials are independent of each other.
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Example 2 Determine whether or not the following examples are Bernoulli experiments.
1. Flipping a coin 6 times and noting the number of times it lands on heads.
2. Flipping a coin and noting the number of flips before you get 6 heads.
3. Picking 5 cards out of a standard 52-card deck, one at a time and with replacement,
and noting the number of hearts.
4. Picking 5 cards out of a standard 52-card deck, without replacement, and noting the
number of hearts.
GOAL: Given a Bernoulli trial repeated n times, we want to find the probability that a
specific number of successes occur.
Notation: P (X = k)
Recall If two events E and F are independent, then the probability that both events
occur, P (E ∩ F ) is given by
P (E ∩ F ) = P (E) · P (F )
Example 3 An unfair coin is to be flipped 5 times. For each flip, the probability of heads
is .7.
a) What is the probability that only the first two flips are heads?
b) What is the probability that the second and fourth flips are heads?
c) What is the probability that exactly two flips are heads?
d) What is the probability that exactly 4 flips are heads?
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If we have a sequence of n repeated Bernoulli Trials, each with probability p of success
and probability q of failure, then the probability of having exactly k successes is given by
Note: We always have p + q = 1. Therefore, if we are only given p, we can always
calculate q.
Computing the Probability of Exactly k Successes in n repeated Bernoulli
Trials (i.e. P (X = k)) using Your Calculator:
(1) Press 2nd.
(2) Press DISTR (above VARS key).
(3) Select binompdf( (Option A).
(4) Enter the number of trials, n; the comma key the probability of success, p; the comma
key, and then the number of successful trials, k,
(5) Press ENTER.
Note: The result will be binompdf(n,p,k)
Example 4 Consider the experiment from Example 3. What is the probability of flipping
2 heads or less?
Computing the Probability of Having Less Than or Equal to k Successes in n
repeated Bernoulli Trials (i.e. P (0 ≤ X ≤ k)) using Your Calculator:
(1) Press 2nd.
(2) Press DISTR (above VARS key).
(3) Select binomcdf( (Option B).
(4) Enter the number of trials, n; the comma key the probability of success, p; the comma
key, and then the maximum number of successful trials, k,
(5) Press ENTER.
Note: The result will be binomcdf(n,p,k)
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Example 5 The coin from the previous examples, with P (H) = 0.7, is now flipped 1000
times.
a) What is the probability that there are at most 690 heads?
b) What is the probability that there are more than 690 heads?
c) What is the probability that there are at least 690 heads?
Example 6 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 brwosing in the store
a) At most five will rent a movie? (Round your answer to 4 decimal places.)
b) At least five will rent a movie? (Round your answer to 4 decimal places.)
c) More than 3 but fewer than 10 will rent a movie? (Round your answer to 4 decimal
places.)
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Example 7 (From Tan) Suppose 30% of the restaurants in a certain part of town are in
violation of the health code. If a health inspector randomly selects 25 of the restaurants
for inspection, what is the probability that
a) None of the restaurants are in violation of the health code? (Round your answer to 4
decimal places.)
b) One of the restaurants are in violation of the health code? (Round your answer to 4
decimal places.)
c) At least 10 of the restaurants are in violation of the health code? (Round your answer
to 4 decimal places.)
Example 8 A final exam has 30 multiple choice questions. Eighteen must be answered
correctly to receive a passing grade. If each question has 5 possible answers (a - e), of
which only one is correct, what is the probability a student guessing at random on each
question will pass the exam? (Round your answer to 4 decimal places)
Example 9 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? (Round your
answer to 4 decimal places)
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