Math Portal
Introductory Statistics
Section 5.1 - The Binomial Distribution
Binomial distributions are important because they allow you to deal with random situations that result in only two relevant outcomes, such as yes or no, pass or fail, defective or nondefective, or correct or incorrect. This section covers this very useful discrete distribution.
Introduction
Consider n independent trials of an experiment where each trial there are exactly two possible outcomes. Outcome 1 is a success with probability p that is constant from trial to trial. Outcome 2 is a failure with a probability of 1 - p (in other words the complement of p).
EXAMPLE: The probability of success for a particular event is .94 or 94%, so the probability for failure would be .06 or 6%
Let X equal the number of successes out of the n repeated trials. The random variable X is called a binomial random variable, and its probability distribution is called a binomial distribution. To summarize, a binomial distribution results from a process that has the following characteristics:
EXAMPLE: Flip a fair coin 20 times and count the number of times heads shows on the up face of the coin.
SOLUTION: The binomial distribution is an appropriate model for the probability distribution of this experiment because (1) there are n identical trials: 20 flips of the coin; (2) each trial (flip) results in only two outcomes: heads or tails, where the outcome heads will denote success; (3) p, the probability of success (heads) on a single trial (flip) remains the same from trial to trial; (4) the trials (flips) are independent since the outcome of one flip of the coin does not affect the outcome of any other flip.
EXAMPLE: Each student in a random sample of 100 students from a certain university is interviewed and asked whether he or she favors construction of a new athletic facility, and the number of yes responses is recorded. Assume that none of the students interviewed refuses to answer the question.
SOLUTION: The binomial distribution is an appropriate model for the probability distribution of this experiment because (1) there are n identical trials: 100 interviews, all the same; (2) each trial (interview) results in only two outcomes: yes or no, where the outcome yes will denote success; (3) p, the probability of success (yes) on a single trial (interview) remains the same (for all practical purposes) from trial to trial; (4) the trials are independent since the outcome of one interview does not affect the outcome of any other interview.
EXAMPLE: Each student in a sample consisting of 50 music majors and 50 kinesiology majors from a certain university is interviewed and asked whether he or she favors construction of a new athletic facility, and the number of yes responses is recorded. Assume that none of the students interviewed refuses to answer the question.
SOLUTION: The binomial distribution is not an appropriate model for the probability distribution of this experiment because the characteristic of independent trials likely would not be satisfied since you can expect that responses of students from the same major will tend to be similar. For instance, most likely the kinesiology majors would tend to be in favor of a new athletic facility.
Binomial distributions are important because they allow you to deal with random situations that result in only two relevant outcomes, such as yes or no, pass or fail, defective or nondefective, or correct or incorrect. This section covers this very useful discrete distribution.
Introduction
Consider n independent trials of an experiment where each trial there are exactly two possible outcomes. Outcome 1 is a success with probability p that is constant from trial to trial. Outcome 2 is a failure with a probability of 1 - p (in other words the complement of p).
EXAMPLE: The probability of success for a particular event is .94 or 94%, so the probability for failure would be .06 or 6%
Let X equal the number of successes out of the n repeated trials. The random variable X is called a binomial random variable, and its probability distribution is called a binomial distribution. To summarize, a binomial distribution results from a process that has the following characteristics:
- There are a fixed number n of identical trials.
- Each trial results in exactly one of only two outcomes, which, by convention, are labeled success and failure.
- The probability of success, denoted p, on a single trial remains the same from trial to trial.
- The trials are independent.
EXAMPLE: Flip a fair coin 20 times and count the number of times heads shows on the up face of the coin.
SOLUTION: The binomial distribution is an appropriate model for the probability distribution of this experiment because (1) there are n identical trials: 20 flips of the coin; (2) each trial (flip) results in only two outcomes: heads or tails, where the outcome heads will denote success; (3) p, the probability of success (heads) on a single trial (flip) remains the same from trial to trial; (4) the trials (flips) are independent since the outcome of one flip of the coin does not affect the outcome of any other flip.
EXAMPLE: Each student in a random sample of 100 students from a certain university is interviewed and asked whether he or she favors construction of a new athletic facility, and the number of yes responses is recorded. Assume that none of the students interviewed refuses to answer the question.
SOLUTION: The binomial distribution is an appropriate model for the probability distribution of this experiment because (1) there are n identical trials: 100 interviews, all the same; (2) each trial (interview) results in only two outcomes: yes or no, where the outcome yes will denote success; (3) p, the probability of success (yes) on a single trial (interview) remains the same (for all practical purposes) from trial to trial; (4) the trials are independent since the outcome of one interview does not affect the outcome of any other interview.
EXAMPLE: Each student in a sample consisting of 50 music majors and 50 kinesiology majors from a certain university is interviewed and asked whether he or she favors construction of a new athletic facility, and the number of yes responses is recorded. Assume that none of the students interviewed refuses to answer the question.
SOLUTION: The binomial distribution is not an appropriate model for the probability distribution of this experiment because the characteristic of independent trials likely would not be satisfied since you can expect that responses of students from the same major will tend to be similar. For instance, most likely the kinesiology majors would tend to be in favor of a new athletic facility.