Binomial Distributions
Binomial Experiment The experiment is repeated for a fixed number of trials, where each trial is independent of the other trials There are only two possible outcomes of interest for each trial. They can be classified as a success or a failure The probability of a success is the same for each trial The random variable x counts the number of successful trials
Notation for Binomial Experiments n The number of times a trial is repeated p The probability of success in a single trial q The probability of failure in a single trial (q p) x The random variable represents a count of the number of successes in n trials
Binomial Experiments Decide whether the experiment is a binomial experiment. If it is specify the values of n, p, q and list the possible random variables x. If it is not explain why. A certain surgical procedure has an 85% chance of success. A doctor performs the procedure on eight patients. The random variable represents the number of successful surgeries. Yes it satisfies the four conditions of a binomial experiment. n 8, p 0.85, q 0.85 0.5, x 0,,2,,4,5,6,7,8,
Binomial Experiments Decide whether the experiment is a binomial experiment. If it is specify the values of n, p, q and list the possible random variables x. If it is not explain why. A jar contains five red marbles, nine blue marbles, and six green marbles. You randomly select three marbles from the jar without replacement. The random variable represents the number of red marbles. No, the trials are not independent of each other, so the probability of success is not the same for each trial
Binomial Experiments Decide whether the experiment is a binomial experiment. If it is specify the values of n, p, q and list the possible random variables x. If it is not explain why. You take a multiple choice quiz that consists of 0 questions. Each question has four possible answers, only one of which is correct. To complete the quiz, you randomly guess the answer to each question. The random variable represents the number of correct answers. Yes it satisfies the four conditions of a binomial experiment. n 0, p 0.25, q 0.75, x 0,,2,,4,5,6,7,8,9,0
Binomial robability Formula ( x) n C x p x q n x n! ( n x)! x! p x q n x
Finding Binomial robabilities A six-sided die is rolled three times. Find the probability of rolling exactly one six. In the binomial experiment rolling a six is a success and rolling anything else is a failure., 6 5, 6, x q p n () ( ) 0.47 6 25 6 6 5 6!!! 6 5 6 2 C
Finding Binomial robabilities A card is selected from a standard deck and replaced. This experiment is repeated five times. Find the probability of selecting exactly clubs. In the binomial experiment selecting a club is a success and selecting anything else is a failure. n 5, p 0.25, q 0.75, x 0 () ( )( C 0.25 0.75) 5 5 ( 0.05625)(.5625) 0. 088 5!!! ( 5 ) ( ) 0.25 ( 0.75) 2
Finding Binomial robabilities Using Formulas A survey indicates that 4% of women in the United States consider reading as their favorite leisure-time activity. You randomly select four U.S. women and ask them if reading if reading is the favorite leisure-time activity. Find the probability that exactly two of them respond yes. n 4, p 0.4, q 0.59, x 2 ( ) ( ) 2( ) 2 2 C 0.4 0.59 0. 5 4 2
Finding Binomial robabilities Using Formulas A survey indicates that 4% of women in the United States consider reading as their favorite leisure-time activity. You randomly select four U.S. women and ask them if reading if reading is the favorite leisure-time activity. Find the probability that at least two of the responded yes. ( x 2) ( 2) + ( ) ( 4) + n 4, p 0.4, q 0.59, x 2,,4 ( ) ( ) 2( ) 2 2 4C2 0.4 0.59 0. 5 () ( )( ) 4C 0.4 0.59 0. 6 ( ) ( ) 4( ) 0 4 C 0.4 0.59 0. 028 4 4 ( x 2) 0.5+ 0.6 + 0.028. 542
Finding Binomial robabilities Using Formulas A survey indicates that 4% of women in the United States consider reading as their favorite leisure-time activity. You randomly select four U.S. women and ask them if reading if reading is the favorite leisure-time activity. Find the probability that fewer than two of them respond yes. ( x < 2) ( 0) ( ) + n 4, p 0.4, q 0.59, x 0, ( ) ( ) 0( ) 4 0 C 0.4 0.59 0. 2 4 0 () ( )( ) C 0.4 0.59 0. 7 4 ( x < 2) 0.2+ 0.7 0. 458
Finding a Binomial robability Using a Table Fifty percent of working adults spend less than 20 minutes each way commuting to their jobs. You randomly select six working adults. What is the probability that exactly three of them spend less than 20 minutes commuting to work? n 6, p 0.5, q 0.5, x Use table 2 in appendix B () 0. 2
Finding a Binomial robability Using a Table Thirty percent of all small businesses in the United States have a website. If you randomly select 0 small businesses, what is the probability that exactly four of the have a website? n 0, p 0., q 0.7, x 4 Use table 2 in appendix B ( 4) 0. 200
Mean, Variance, and Standard Deviation Mean : µ np Variance: 2 σ npq Standard Deviation : σ npq
Finding Mean, Variance, and Standard Deviation In ittsburgh, ennsylvania, about 56% of the days in the year are cloudy. Find the mean, variance, and standard deviation for the number of cloudy days during the month of June. n 0, p 0.56, q 0.44 Mean : µ 2 Variance: σ Standard Deviation : σ np 0 0.56 6.8 npq 0 0.56 0.44 7.92 npq 7.92 2.79