OPMT 1197
Business Statistics
Lecture 7/8: Binomial Probability Distribution
Characteristics of the Binomial Distribution:
1. There are a fixed number of identical trials. Each trial has only one of two possible mutually
exclusive outcomes, often referred to as “success” and “failure”.
2. Trial outcomes are independent. The outcome of one does not affect the outcome of another.
3. Results from counting something → Counting the number of successes.
4. The probability of success is the same for each trial.
1. A quiz consists of 5 multiple-choice questions with 4 possible answers for each question. The
student is unprepared for the exam and randomly selects answers. Verify that this example
possesses the 4 characteristics of a binomial. Find the probability that the student gets:
(a) all five questions correct? (b) no correct answers?
(c) only the first 2 questions correct? (ii) only the 1st and 3rd questions correct?
(iii) In how many different ways can you get exactly 2 correct answers in 5 questions?
(iv) Find the probability that the student correctly answers exactly two questions.
Now use the Binomial Formula to find the probability that the student gets:
(a) all five correct (b) none correct (c) exactly two correct (d) at least one correct
(e) at least two correct (f) at least four correct (g) at most four correct
2. A student wrote a quiz consisting of 8 multiple-choice questions with five possible answers
for each question. The student wrote Version 2 of the quiz but had all the correct answers for
Version 1. The student claimed that he randomly guessed answers to all the questions. What
is the probability he guessed all the correct answers to Version 1?
3. Van Air offers several direct flights from Vancouver to Victoria. Van Air has a policy of
overbooking their planes. Past experience has shown that only 90% of the passengers who
purchase a ticket actually show up for the flight. If too many passengers show up for the
flight, Van Air will ask for a volunteer to give up their seat in exchange for a free ticket.
11 passengers have purchased tickets on a flight that has only 10 seats.
(a) What is the probability of the flight being exactly 80% full?
(b) What is the probability that there are enough seats so that every passenger who shows up
will get a seat on the plane?
(c) What is the probability there will be at least one empty seat? (i.e. the flight is not full)
(d) You and your partner show up without a reservation and ask to go standby. What is the
probability that the two of you will get a seat on this flight?
(e) What is the probability of at most two passengers not showing up for the flight?
4. TV World anticipates selling forty large-screen television sets this month. Typically, about
30% of customers buy an extended warranty which sells for $250. If they sell 40 large-screen
televisions, what is the expected revenue from the sale of the warranties?
Sol: 1. (a) 0.000977 (b) 0.2373 (c) 0.2637 (d) 0.7627 (e) 0.3672 (f) 0.0156 (g) 0.9990
2. 0.00000256 or only a 1/390,625 chance.
3. (a) 0.0710 (b) 0.6862 (c) 0.3026 (d) 0.0896 (e) 0.9104 4. $3,000
Business Statistics
Lecture 7/8: Binomial Probability Distribution
Characteristics of the Binomial Distribution:
1. There are a fixed number of identical trials. Each trial has only one of two possible mutually
exclusive outcomes, often referred to as “success” and “failure”.
2. Trial outcomes are independent. The outcome of one does not affect the outcome of another.
3. Results from counting something → Counting the number of successes.
4. The probability of success is the same for each trial.
1. A quiz consists of 5 multiple-choice questions with 4 possible answers for each question. The
student is unprepared for the exam and randomly selects answers. Verify that this example
possesses the 4 characteristics of a binomial. Find the probability that the student gets:
(a) all five questions correct? (b) no correct answers?
(c) only the first 2 questions correct? (ii) only the 1st and 3rd questions correct?
(iii) In how many different ways can you get exactly 2 correct answers in 5 questions?
(iv) Find the probability that the student correctly answers exactly two questions.
Now use the Binomial Formula to find the probability that the student gets:
(a) all five correct (b) none correct (c) exactly two correct (d) at least one correct
(e) at least two correct (f) at least four correct (g) at most four correct
2. A student wrote a quiz consisting of 8 multiple-choice questions with five possible answers
for each question. The student wrote Version 2 of the quiz but had all the correct answers for
Version 1. The student claimed that he randomly guessed answers to all the questions. What
is the probability he guessed all the correct answers to Version 1?
3. Van Air offers several direct flights from Vancouver to Victoria. Van Air has a policy of
overbooking their planes. Past experience has shown that only 90% of the passengers who
purchase a ticket actually show up for the flight. If too many passengers show up for the
flight, Van Air will ask for a volunteer to give up their seat in exchange for a free ticket.
11 passengers have purchased tickets on a flight that has only 10 seats.
(a) What is the probability of the flight being exactly 80% full?
(b) What is the probability that there are enough seats so that every passenger who shows up
will get a seat on the plane?
(c) What is the probability there will be at least one empty seat? (i.e. the flight is not full)
(d) You and your partner show up without a reservation and ask to go standby. What is the
probability that the two of you will get a seat on this flight?
(e) What is the probability of at most two passengers not showing up for the flight?
4. TV World anticipates selling forty large-screen television sets this month. Typically, about
30% of customers buy an extended warranty which sells for $250. If they sell 40 large-screen
televisions, what is the expected revenue from the sale of the warranties?
Sol: 1. (a) 0.000977 (b) 0.2373 (c) 0.2637 (d) 0.7627 (e) 0.3672 (f) 0.0156 (g) 0.9990
2. 0.00000256 or only a 1/390,625 chance.
3. (a) 0.0710 (b) 0.6862 (c) 0.3026 (d) 0.0896 (e) 0.9104 4. $3,000
