The Binomial distribution Examples and Definition Binomial Model an experiment 1 A series of n independent trials is conducted 2 Each trial results in a binary outcome one is labeled success the other failure 3 The probability of success is equal to p for each trial regardless of the outcomes of the other trials Binomial Random Variable The number of successes in the binomial experiment Let Y of success in the above model Then Y is a binomial random variable with parameters n sample size and p success probability It is often denoted Y B n p Example Tossing a fair coin Toss a fair coin three times P H 5 Interest is counting the number of heads Y of heads success heads failure tails n p Y Example Tossing a fair coin Toss a fair coin three times P H 5 Interest is counting the number of heads Y of heads success heads failure tails n 3 p Y Example Tossing a fair coin Toss a fair coin three times P H 5 Interest is counting the number of heads Y of heads success heads failure tails n 3 p 5 Y Example Tossing a fair coin Toss a fair coin three times P H 5 Interest is counting the number of heads Y of heads success heads failure tails n 3 p 5 Y B 3 5 Example Tossing an unfair coin Toss a biased coin 5 times P H 7 Interest is counting the number of heads Y of heads success heads failure tails n 5 p 7 Y B 5 7 Example Counting Mutations Experiment to mutate a gene in bacteria the probability of causing a mutation is 4 The experiment was repeated 10 times with 10 independent colonies Interest is counting the number of mutations Y of mutations success mutation failure no mutation n 10 p 4 Y B 10 4 Computing Probabilities for a Binomial Random Variable Board Example Tossing a biased coin Y B 3 7 Tossing a fair Coin Consider tossing a fair coin 3 times Y of heads in the 3 tosses Y B 3 5 Consider the Possible outcomes TTT HTT THT TTH HHT HTH THH HHH 1 1 1 1 2 2 2 8 1 1 1 3 IP Y 1 3 2 2 2 8 1 1 1 3 IP Y 2 3 2 2 2 8 1 1 1 1 IP Y 3 2 2 2 8 IP Y 0 Tossing a Biased Coin Consider tossing a biased coin 3 times IP H 7 Y of heads in the 3 tosses Y B 3 7 Consider the Possible outcomes TTT HTT THT TTH HHT HTH THH HHH IP Y 0 1 3 3 3 1 70 33 027 IP Y 1 IP Y 2 IP Y 3 Tossing a Biased Coin Consider tossing a biased coin 3 times IP H 7 Y of heads in the 3 tosses Y B 3 7 Consider the Possible outcomes TTT HTT THT TTH HHT HTH THH HHH IP Y 0 1 3 3 3 1 70 33 027 IP Y 1 3 7 3 3 3 71 32 189 IP Y 2 IP Y 3 Tossing a Biased Coin Consider tossing a biased coin 3 times IP H 7 Y of heads in the 3 tosses Y B 3 7 Consider the Possible outcomes TTT HTT THT TTH HHT HTH THH HHH IP Y 0 1 3 3 3 1 70 33 027 IP Y 1 3 7 3 3 3 71 32 189 IP Y 2 3 7 7 3 3 72 31 441 IP Y 3 Tossing a Biased Coin Consider tossing a biased coin 3 times IP H 7 Y of heads in the 3 tosses Y B 3 7 Consider the Possible outcomes TTT HTT THT TTH HHT HTH THH HHH IP Y 0 1 3 3 3 1 70 33 027 IP Y 1 3 7 3 3 3 71 32 189 IP Y 2 3 7 7 3 3 72 31 441 IP Y 3 1 7 7 7 1 73 30 343 Question Is there a general formula for computing Binomial probabilities We do not want to have to list all possibilities when n 10 or n 100 Background Factorials Factorial Multiply all numbers from 1 to n n is a positive integer n n n 1 n 2 2 1 Example n 4 n 4 3 2 1 24 Note 0 1 Background Binomial Coefficients n Cj n Cj n j n j n Cj counts all of the ways that j successes can appear in n total trials Example n 5 j 3 5 4 5 10 3 2 2 cf Table 2 p 674 for the n Cj numbers Chalkboard Y B n p Can we guess the answer IP Y j Binomial Distribution Formula IP Y j n Cj pj 1 p n j Y B n p a binomial random variable based on n trials and success probability p The probability of getting j successes is given by j 0 1 n IP Y j pj 1 p n j n pj 1 p n j j n j n Cj Example IP Y j n pj 1 p n j j n j Question You toss a fair coin 3 times what is the probability of getting 2 heads Answer p 5 fair coin n 3 tosses j 2 Heads We get IP Y 2 3 3 2 1 5 2 5 1 5 3 3 8 2 1 2 1 1 Example IP Y j n pj 1 p n j j n j Question Y B 7 6 Compute IP Y 2 Answer p 6 n 7 tosses j 2 Heads We get 7 7 6 5 6 2 4 5 6 2 4 5 2 5 1 2 5 21 6 2 4 5 0774 IP Y 2 Example A new drug is available Its success rate is 1 6 probability that a patient is improved I try it independently on 6 patients Probability that at least one patient improves p 1 6 n 6 j 1 2 3 4 5 or 6 IP at least one improves IP Y 1 or Y 2 or or Y 6 IP Y 1 IP Y 2 IP Y 6 1 IP Y 0 6 1 1 6 0 5 6 6 1 5 6 6 0 6 665 Probability Distribution IP Y j n pj 1 p n j j n j Y B 6 1 6 n 6 p 1 6 We can compute IP Y j for j 0 1 2 3 4 5 6 y IP Y y 0 0 335 1 0 402 2 0 200 3 0 054 4 0 008 5 0 0006 6 0 00002 Mean Variance and Standard Deviation for Binomial Random Variables Recall the Formula for Mean Variance for a General Discrete Random Variable X E Y yi IP Y yi X Var Y yi Y 2 IP Y yi where the yi s are the values that the variable takes on and the sum is taken …
View Full Document
Unlocking...