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Math 1313 Section 8. Binomial Distribution
We will finish the course by looking at two common distributions of random variables, the binomial distribution and the normal distribution. In this section, we’ll take a look at the binomial distribution.
Definition: A binomial trial (or a Bernoulli trial) is an experiment which has exactly two outcomes, which we can label as success and failure.
Definition: A binomial experiment is a sequence of binomial trials.
Binomial experiments have these properties in common:
- the number of trials is fixed
- there are 2 outcomes, success and failure
- the probability of success is the same in each trial
- the trials are independent (i.e., one doesn’t depend on the outcome of another)
Example 1 : A fair die is cast four times. Find the probability of obtaining exactly one “three” in the four trials. Is this a binomial experiment?
We use some standard notation in working with binomial experiments:
p = probability of success q = probability of failure p + q = 1, so q = 1 – p and p = 1 – q n = number of trials
In a binomial experiment, we can define a random variable X to be the number of successes in n independent trials.
Computing the Probability of x Success in n Trials of a Binomial Experiment
We can compute the probability of x successes in n independent trials of a binomial experiment, with p = probability of success and q = probability of failure by
P X ( = x ) = C n x p q ( , ) x^ n^ − x.
Example 2 : A fair die is cast four times. Find the probability of obtaining exactly one “three” in the four trials.
Example 3 : Consider the following binomial experiment. The probability that a randomly selected student at a certain college will graduate with a bachelor’s degree after four years of study is .78. From among a group of 15 students at this college, what is the probability that
a. all of them will graduate after four years of study?
b. exactly 10 of them will graduate after four years?
c. At least one graduate after four years.
We compute the mean, variance and standard deviation of binomial probability distributions using different formulas from the ones we have already learned.
Example 6. In a certain congressional district, it is know that 40% of the registered voters classify themselves as conservatives. If ten registered voters are selected at random from the district, what is the probability that four of them will say they are conservatives?
Example 7 : Six newly married couples agree to be part of a 20 year survey. Studies show that the probability that a marriage will end in divorce is .6 within 20 years of its start. What is the probability that, out of the 6 surveyed couples, at the end of 20 years a. none will be divorced?
b. all will be divorced?
c. at least two couples will be divorced?
Example 8 : State the mean, variance and standard deviation of the binomial experiment state in Example 7.
Example 9: The Krazy Toy Company makes toy phones. The quality control department estimates that 5% of the phones made are defective. A random sample of 20 phones is made from a large shipment of toy phones. What is the probability that the sample contains at most 2 defective phones?
What is the probability that at least one phone is defective?
