Teacher Introductory Statistics Lesson 4.2 B Objective: SSBAT find binomial probabilities using the binomial probability formula. Standard: S2.5B

Binomial Probability Formula Review Binomial Probability Formula The probability of exactly x successes in n trials is: P(x) = nCx px qn-x Where: n = Number of times a trial is Repeated p = Probability of Success P(S) q = Probability of Failure P(F) x = Represents a count of the number of successes in n trials: x = 0, 1, 2, 3, …, n

Probability of an OR statement  Add the probabilities Review: Probability of an OR statement  Add the probabilities P(Rolling a 2 or 3) = P(2) + P(3) P(Rolling at least a 4) = P(4) + P(5) + P(6) P(Rolling a number < 3) = P(1) + P(2)

P(at least 2) = P(2) + P(3) + P(4) Examples: A survey indicates that 41% of women in the U.S. consider reading their favorite leisure-time activity. You randomly select 4 U.S. women. Find the probability that at least 2 of them said reading is their favorite leisure-time activity. n = 4, p = .41, q = 1 – 0.41 = 0.59, x = 2, 3, or 4 P(at least 2) = P(2) + P(3) + P(4)  Use the formula to find P(2), P(3), and P(4)  Then add the 3 probabilities together P(x) = nCx px qn-x

Teacher P(x) = nCx px qn-x #1 Continued. n = 4, p = .41, q = 1 – 0.41 = 0.59, x = 2, 3, or 4 P(2) + P(3) + P(4) P(2) = 4C2 (.41)2 (.59)2 P(3) = 4C3 (.41)3 (.59)1 P(4) = 4C4 (.41)4 (.59)0 = 0.351 = 0.163 = 0.028 P(at least 2) = 0.351 + 0.163 + 0.028 P(at least 2) = 0.542

P(fewer than 2) = P(0) + P(1) P(x) = nCx px qn-x A survey indicates that 21% of men in the U.S. consider fishing their favorite leisure-time activity. You randomly select 5 U.S. men. Find the probability that fewer than 2 of them said fishing is their favorite leisure-time activity. n = 5, p = .21, q = 1 – 0.21 = 0.79, x = 0 or 1 P(fewer than 2) = P(0) + P(1)  Use the formula to find P(0) and P(1)  Then add the 2 probabilities together

P(0) = 5C0 (.21)0 (.79)5 P(1) = 5C1 (.21)1 (.79)4 = 0.308 = 0.409 Teacher P(x) = nCx px qn-x #2 Continued. n = 5, p = .21, q = 0.79, x = 0 or 1 P(0) + P(1) P(0) = 5C0 (.21)0 (.79)5 P(1) = 5C1 (.21)1 (.79)4 = 0.308 = 0.409 P(fewer than 2) = 0.308 + 0.409 P(fewer than 2) = 0.717

P(4 or more) = P(4) + P(5) + P(6) + P(7) P(x) = nCx px qn-x Micro-fracture knee surgery has a 75% chance of success on patients with degenerative knees. The surgery is performed on 7 patients. Find the probability that 4 or more patients will have success. n = 7, p = .75, q = 1 – 0.75 = 0.25, x = 4, 5, 6, or 7 P(4 or more) = P(4) + P(5) + P(6) + P(7)

Teacher #3 Continued. n = 7, p = .75, q = 0.25, x = 4, 5, 6, 7 P(4) + P(5) + P(6) + P(7) P(4) = 7C4 (.75)4 (.25)3 P(5) = 7C5 (.75)5 (.25)2 P(6) = 7C6 (.75)6 (.25)1 P(7) = 7C7 (.75)7 (.25)0 P(x) = nCx px qn-x = 0.173 = 0.311 = 0.311 = 0.133 P(4 or more) = 0.173 + 0.311 + 0.311 + 0.133 P(4 or more) = 0.928

P(at most 2) = P(0) + P(1) + P(2) P(x) = nCx px qn-x You are taking a multiple-choice quiz that consists of 5 questions. Each question has 4 possible answers, only one of which is correct. To complete the quiz, you randomly guess the answer to each questions. Find the probability of guessing at most 2 questions correctly. n = 5, p = ¼ = 0.25, q = 1 – 0.25 = 0.75, x = 0, 1, 2 P(at most 2) = P(0) + P(1) + P(2)

Teacher #4 Continued. n = 5, p = .25, q = 0.75, x = 0, 1, 2 P(0) + P(1) + P(2) P(0) = 5C0 (.25)0 (.75)5 P(1) = 5C1 (.25)1 (.75)4 P(2) = 5C2 (.25)2 (.75)3 P(x) = nCx px qn-x = 0.237 = 0.396 = 0.264 P(at most 2) = 0.237 + 0.396 + 0.264 P(at most 2) = 0.897

Complete Worksheet 4.2B