The Binomial Distribution

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Presentation transcript:

The Binomial Distribution Chapter 5.3 The Binomial Distribution

Binomial Experiment Probability problems that have only two outcomes, or can be reduced to two outcomes, are called binomial experiments. For example: flipping a coin, gender of a baby, win or loss, true/false, effective or ineffective, normal or abnormal, correct or incorrect

Requirements of a binomial experiment There must be a fixed number of trials Each trial can have only two outcomes. These outcomes can be considered as either success or failure. The outcomes of each trial must be independent of one another The probability of a success must remain the same for each trial.

Notation for a Binomial Distribution P(S) – the symbol for the probability of success P(F) – the symbol for the probability of failure p – the numerical probability of a success q – the numerical probability of a failure n – the number of trials X – the number of success in n trials So, P(S) = p and P(F) = 1 – p = q

The binomial probability formula In a binomial experiment, the probability of exactly X successes in n trials is

Why does it work? Tossing coins… A coin is tossed three times. Find the probability of getting exactly two heads.

Survey on Employment A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part-time jobs. If five teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs.

Survey on Doctor Visits A survey found that one out of five Americans say he or she has visited a doctor in any given month. If ten people are selected at random, find the probability that exactly three will have visited a doctor last month.

Survey on Fear of Being Home Alone at Night Public Opinion reported that 5% of Americans are afraid of being alone in a house at night. If a random sample of 20 Americans is selected, find these probabilities. There are exactly five people in the sample who are afraid of being alone at night The are at most three people in the sample who are afraid of being alone at night There are at least three people in the sample who are afraid of being alone at night

Driving While Intoxicated A report from the Secretary of Health and Human Services stated that 70% of single-vehicle traffic fatalities that occur at night on weekends involve an intoxicated driver. If a sample of 15 single-vehicle traffic fatalities that occur at night on a weekend is selected, find the probability that exactly 12 involve a driver who is intoxicated.

Mean, Variance, and Standard Deviation for the Binomial Distribution

Tossing a Coin A coin is tossed four times. Find the mean, variance, and standard deviation of the number of heads that will be obtained.

Rolling a Die A die is rolled 480 times. Find the mean, variance, and standard deviation of the number of “3’s” that will be rolled.

Likelihood of Twins The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of all American births result in twins. If a random sample of 8000 births is taken, find the mean, variance, and standard deviation of the number of births that would result in twins.