Math 1680 Spring 2008Page 1 of 11
Chapters 15
Chapter 15: The Binomial Formula
Certain probabilities bear a resemblance to each other. Consider the following questions:
Ex. #1: A parolee has a 24% chance of becoming a repeat offender independent of other parolees. What is the probability that exactly two of five parolees will become repeat offenders?
Ex. #2: A roulette wheel is spun 20 times. What is the probability that the ball will land in a black slot at least 11 times?
Ex. #3: A fair coin is flipped 400 times. What is the probability that it will land heads 220 times or more?
Binomial Experiments
These are all examples of binomial experiments, which are characterized by the following:
1. There are a fixed number of trials. We call this number .
2. The trials are independent and are repeated under identical conditions.
3. Each trial has only two outcomes – success (S) and failure (F)
4. For each trial, the probability of success is the same. We denote the probability of success by and the probability of failure by . Obviously, .
5. The central problem is to find the probability of r successes out of n trials.
Solution #1: Parolees - (25.3%)
1.
2. Independence is assumed for the parolees.
3. S= becomes a repeat offender
F= does not become a repeat offender
4. p = 0.24
q = 1 – p = 1 – 0.24 = 0.76
5. We seek the probability of two successes out of five independent trials.
Solution #2: Roulette - (32.2%)
1.
2. We assume that the wheel is not rigged
3. S = lands black
F = does not land black
4.
5. We seek the probability of at least 11 successes out of 20 independent trials.
Solution #3: Coin Flips - (2.5%)
1.
2. We assume the coin is fair.
3. S = heads
F = tails
4.
5. We seek the probability of at least 220 successes out of 400 independent trials.
Ex. #4: Explain why the following are NOT binomial experiments.
A) A company manager has ten employees – six females, four males. Two are selected at random to attend a conference. What is the probability that both are females.
B) The students in this class are asked, "What is their favorite TV show?"
The Binomial Formula – Derivation
Ex. #5: A student takes a multiple-choice exam, where each question has five possible answers. At the end of the exam, she answers all questions except for three, for which she picks answers randomly. What is the probability that she got all three questions correct? Two of the three correct? One of the three correct? None of the guesses correct?
Solution: Notice that this is a binomial experiment:
1. There are three trials – questions to answer. So .
2. The trials are independent.
3. S = correct answer
F = incorrect answer
4. For each trial, and
5. The central problem is determining the probability of 0, 1, 2, or 3 successes.
Method #1: Calculate each possible outcome individually and compute appropriately. (Without a more efficient method of calculating this probability, we must compute it this way.)
First, notice there are eight possible outcomes:
Next, find the probabilities of each of these occurring.
Summarizing,
/ - probability of successes in 3 trials / for0 / / 0.512
1 / / 0.384
2 / / 0.096
3 / / 0.008
Not surprisingly, ______.
Note: The probability of one success is not , since there is more than one way to get a single success.
The Binomial Formula– The chance that an event will occur exactly times out of trials is given by the binomial formula:
Definition: Factorial. The factorial, denoted , where is a positive integer is defined by:
Ex. #6:
Most calculators have a key for computing . However, even if your calculator doesn’t, you can find these with a little multiplication and division:
Note: There is no such thing as -- you can’t have 4 successes in 3 trials.
Tip: You can use either the formula or Pascal’s triangle to find .
Ex. #7: A student randomly guesses at three questions. Each question has five possible answers, only once of which is correct. Find the probability that she gets 0, 1, 2 or 3 correct. Same problem as the previous one, we will solve this one by means of the binomial formula.
Solution: In this example, ______; ______; and .
So,
Ex. #8: A die is rolled 12 times. What is the probability of getting exactly two aces?
Solution: ______;______; success = ______;
______; and
Now, using:
Ex. #9: A survey claims that 70% of students oppose the construction of a new parking lot. A random sample of 20 students is asked about this project.
A) Find the probability that 16 students oppose the project.
B) Find the probability that 14 students oppose the project.
Ex. #10: About 10% of businessmen tie their neckties too tight, reducing blood flow to the brain. At a board meeting of 10 businessmen (all wearing neckties), what is the probability that
A) At least one tie is too tight
B) Exactly two are too tight
C) At least 18 are not too tight
D) Less than three are two tight
Ex. #11: Four draws are made with replacement from a box containing 6 tickets; two of which are labeled “1”, and one each labeled, “2”, “3”, “4” and “5”. Find the probability of getting two “1”s.
Ex. #12: What if the draws are made without replacement?