Binomial Distributions

Binomial Experiments Probability experiments for which the results of each trial can be reduced to two outcomes: success and failure. When a basketball player attempts a free throw, he or she either makes the basket or does not. Probability experiments such as these are called binomial experiments.

Binomial Probability Distribution A Fixed Number of Observations (trials), n 15 tosses of a coin; 20 patients; 1000 people surveyed A Binary Outcome Head or tail in each toss of a coin; disease or no disease Generally called “success” and “failure” Probability of success is p, probability of failure is 1 – p Constant Probability for each observation Probability of getting a tail is the same each time we toss the coin

Notation for Binomial Experiments Symbol Description n The number of times a trial is repeated. p = P(S) The probability of success in a single trial. q = P(F) The probability of failure in a single trial (q = 1 – p) X The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, . . . n.

Example We pick a card from a standard deck of cards, and note whether it is a club or not, and replace the card. We repeat the experiment 5 times. n = 5 p = P(S) = ¼ q = P(F) = ¾ Possible values of the random variable are 0, 1, 2, 3, 4, and 5.

Binomial Experiments Decide whether the experiment is a binomial experiment: A binomial experiment specify the values of n (number of times a trial is repeated), p (Probability of Success), q (Probability of Failure) and list the possible values of the random variable, x.

Example 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.

Binomial Experiment p = 0.85 q = 1 – 0.85 = 0.15 Each surgery represents one trial. There are eight surgeries, and each surgery is independent of the others. Only two possible outcomes for each surgery - either the surgery is a success or it is a failure. n = 8 p = 0.85 q = 1 – 0.85 = 0.15 x = 0, 1, 2, 3, 4, 5, 6, 7, 8

Example A jar contains five red marbles, nine blue marbles and six green marbles. Select randomly three marbles from the jar, without replacement. The random variable represents the number of red marbles.

Not A Binomial Experiment Each marble selection represents one trial and selecting a red marble is a success. When selecting the first marble, the probability of success is 5/20. However because the marble is not replaced, the probability of further trials is no longer 5/20. Trials are not independent.

Binomial Probability Formula

Example: Binomial Probabilities A six sided die is rolled 3 times. Find the probability of rolling exactly one 6.

Roll 1 Roll 2 Roll 3 Frequency # of 6’s Probability (1)(1)(1) = 1 3 1/216 (1)(1)(5) = 5 2 5/216 (1)(5)(1) = 5 (1)(5)(5) = 25 1 25/216 (5)(1)(1) = 5 (5)(1)(5) = 25 (5)(5)(1) = 25 (5)(5)(5) = 125 125/216

Example: Binomial Probabilities Three outcomes that have exactly one six Each has a probability of 25/216 Probability of rolling exactly one six is 3(25/216) ≈ 0.347. Binomial Probability Formula (n = 3, p = 1/6, q = 5/6 and x = 1). Probability of rolling exactly one 6 is: Binomial Probability Formula

Example: Binomial Probabilities

Binomial Probability Distribution By listing the possible values of x with the corresponding probability of each, we can construct a Binomial Probability Distribution.

Constructing a Binomial Distribution In a survey, a company asked their workers and retirees to name their expected sources of retirement income. Seven workers who participated in the survey were asked whether they expect to rely on Pension for retirement income. 36% of the workers responded that they rely on Pension only. Create a binomial probability distribution.

Constructing a Binomial Distribution x P(x) 0.044 1 0.173 2 0.292 3 0.274 4 0.154 5 0.052 6 0.010 7 0.001 P(x) = 1 Notice all the probabilities are between 0 and 1 and that the sum of the probabilities is 1.

Finding a Binomial Probability Using a Table Fifty percent of working adults spend less than 20 minutes commuting to their jobs. If you randomly select six working adults, what is the probability that exactly three of them spend less than 20 minutes commuting to work? Using the distribution for n = 6 and p = 0.5, we can find the probability that x = 3

Population Parameters of a Binomial Distribution Mean:  = np Variance: 2 = npq Standard Deviation:  = √npq

Example In Murree, 57% of the days in a year are cloudy. Find the mean, variance, and standard deviation for the number of cloudy days during the month of June. Mean:  = np = 30(0.57) = 17.1 Variance: 2 = npq = 30(0.57)(0.43) = 7.353 Standard Deviation:  = √npq = √7.353 ≈2.71

Problem 1 Four fair coins are tossed simultaneously. Find the probability function of the random variable X = Number of Heads and compute the probabilities of obtaining no heads, precisely 1 head, at least 1 head, not more than 3 heads.

Problem 2 If the Probability of hitting a target in a single shot is 10% and 10 shots are fired independently. What is the probability that the target will be hit at least once?

Problem 3 If the Probability of hitting a target in a single shot is 5% and 20 shots are fired independently. What is the probability that the target will be hit at least once?

Problem 5 Let X be the number of cars per minute passing a certain point of some road between 8 A.M and 10 A.M on a Sunday. Assume that X has a Poisson distribution with mean 5. Find the probability of observing 3 or fewer cars during any given minute.

Problem 7 In 1910, E. Rutherford and H. Geiger showed experimentally that number of alpha particles emitted per second in a radioactive process is random variable X having a Poisson distribution. If X has mean 0.5. What is the probability of observing 2 or more particles during any given second?

Problem 9 Suppose that in the production of 50л resistors, non-defective items are those that have a resistance between 45л and 55л and the probability of being defective is 0.2%. The resistors are sold in a lot of 100, with the guarantee that all resistors are non-defective. What is the probability that a given lot will violate this guarantee?

Problem 11 Let P = 1% be the probability that a certain type of light bulb will fail in 24 hours test. Find the probability that a sign consisting of 100 such bulbs will burn 24 hours with no bulb failures.

Problem 13 Suppose that a test for extrasensory perception consists of naming (in any order) 3 card randomly drawn from a deck of 13 cards. Find the probability that by chance alone, the person will correctly name (a) no cards, (b) 1 Card, (c) 2 Cards, and (d) 3 cards.

Ex. 6: Finding Binomial Probabilities A survey indicates that 41% of American women consider reading as their favorite leisure time activity. You randomly select four women and ask them if reading is their favorite leisure-time activity. Find the probability that (1) exactly two of them respond yes, (2) at least two of them respond yes, and (3) fewer than two of them respond yes.

Ex. 6: Finding Binomial Probabilities #1--Using n = 4, p = 0.41, q = 0.59 and x =2, the probability that exactly two women will respond yes is: Calculator or look it up on pg. A10

Ex. 6: Finding Binomial Probabilities #2--To find the probability that at least two women will respond yes, you can find the sum of P(2), P(3), and P(4). Using n = 4, p = 0.41, q = 0.59 and x =2, the probability that at least two women will respond yes is: Calculator or look it up on pg. A10

Ex. 6: Finding Binomial Probabilities #3--To find the probability that fewer than two women will respond yes, you can find the sum of P(0) and P(1). Using n = 4, p = 0.41, q = 0.59 and x =2, the probability that at least two women will respond yes is: Calculator or look it up on pg. A10

Ex. 7: Constructing and Graphing a Binomial Distribution 65% of American households subscribe to cable TV. You randomly select six households and ask each if they subscribe to cable TV. Construct a probability distribution for the random variable, x. Then graph the distribution. Calculator or look it up on pg. A10

Ex. 7: Constructing and Graphing a Binomial Distribution 65% of American households subscribe to cable TV. You randomly select six households and ask each if they subscribe to cable TV. Construct a probability distribution for the random variable, x. Then graph the distribution. x 1 2 3 4 5 6 P(x) 0.002 0.020 0.095 0.235 0.328 0.244 0.075 Because each probability is a relative frequency, you can graph the probability using a relative frequency histogram as shown on the next slide.

Ex. 7: Constructing and Graphing a Binomial Distribution Then graph the distribution. x 1 2 3 4 5 6 P(x) 0.002 0.020 0.095 0.235 0.328 0.244 0.075 Relative Frequency NOTE: that the histogram is skewed left. The graph of a binomial distribution with p > .05 is skewed left, while the graph of a binomial distribution with p < .05 is skewed right. The graph of a binomial distribution with p = .05 is symmetric. Households

Mean, Variance and Standard Deviation Although you can use the formulas learned in 4.1 for mean, variance and standard deviation of a probability distribution, the properties of a binomial distribution enable you to use much simpler formulas. They are on the next slide.

Quiz # 3 32 CE(B) – 12 NOV 2012 Let P = 1% be the probability that a certain type of light bulb will fail in 24 hours test. Find the probability that a sign consisting of 10 such bulbs will burn 24 hours with no bulb failures. (3 Marks) Write Probability Distribution Function for Multinomial and Hypergeometric distributions. (2 Marks)