Unit 7 B The At Least Once Rule

Unit 7 B The At Least Once Rule Slide 7 -1

7 -B 100% Probability - Recall P(event A occurring) + P(event A not occurring ) = 100% so P(event A occurring ) = 100% - P(event A not occurring) or P(event A not occurring) = 100% - P(event A occurring) Slide 7 -2

7 -B The At Least Once Rule (For Independent Events) (Not in ALEKS) • Suppose you toss a coin four times. What is the probability of getting at least one head? P(at least one H in 4 tosses) = P(1 H) + P(2 H) + P(3 H) +P(4 H) or P( at least one H in 4 tosses) = 1 – P(no H in 4 tosses) = P(no H) x P(no H) = [P(no H)] 4 = [1/2]4 P( at least one H in 4 tosses) = 1 – P(no H in 4 tosses) = 1 – [P(no H)] 4 = 1 – [1/2]4 = 1 – 1/16 = 15/16

The At Least Once Rule (For Independent Events) 7 -B Suppose the probability of an event A occurring in one trial is P(A). If all trials are independent, the probability that event A occurs at least once in n trials is shown below. P(at least one event A in n trials) = 1 – P(no event A in n trials) = 1 – [P(no A in one trial)]n Slide 7 -4

Example: The At Least Once Rule (For Independent Events) 7 -B You roll a die five times. What is the probability you roll at least one 3? P(at least one 3 in 5 rolls) = 1 – P(no 3 in 5 rolls) P(no 3 in one roll) = 1 – 1/6 = 5/6 P(no 3 in 5 rolls) = (5/6)5 P(at least one 3 in 5 rolls) = 1 – P(no 3 in 5 rolls) = 1 – (5/6)5 = 1 – 3125/7776 = 4651/7776 which is about 0. 5981

Example: The At Least Once Rule (For Independent Events) 7 -B You purchase five lottery tickets for which the probability of winning some prize on a single ticket is 1 in 10. What is the probability you will have at least one winning ticket among the five tickets? P(at least one winner in 5 tickets) = 1 – P(no winner in 5 tickets) P(no winner in a single ticket) = 1 – 0. 1 = 0. 9 P(no winner in 5 tickets) = (0. 9)5 P(at least one winner in 5 tickets) = 1 – P(no winner in 5 tickets) = 1 – (0. 9)5 = 0. 40951, which is about 0. 410