Chapter 17 Part 1 The Geometric Model

𝑃 𝑛𝑜𝑡 𝐵𝑟 𝑎𝑛𝑑 𝑛𝑜𝑡 𝐵𝑟 𝑎𝑛𝑑 𝑛𝑜𝑡 𝐵𝑟 𝑎𝑛𝑑 𝐵𝑟 = 0.9(0.9)(0.9)(0.1) Recall problem from Chapter 14: 10% of M&M’s are brown, 10% are red, 20% are yellow, 20% are blue, 20% are orange, and 20% are green. If you pick four M&M’s in a row, what is the probability that the 4th one is the first one that is brown? 𝑃 𝑛𝑜𝑡 𝐵𝑟 𝑎𝑛𝑑 𝑛𝑜𝑡 𝐵𝑟 𝑎𝑛𝑑 𝑛𝑜𝑡 𝐵𝑟 𝑎𝑛𝑑 𝐵𝑟 = 0.9(0.9)(0.9)(0.1)

Bernoulli Trials There are only two outcomes: success or failure The probability of success is the same for every trial The trials are independent

The 10% Condition If trials are not independent, it is okay to proceed if the sample is less than 10% of the population.

Do these situations involve Bernoulli trials? We roll 50 dice to find the distribution of the number of spots on the faces. How likely is it that in a group of 120 the majority may have Type A blood, given that Type A is found in 43% of the population? We deal 7 cards from a deck and get all hearts. How likely is that? We wish to predict the outcome of a vote on the school budget, and poll 500 of 3000 likely voters to see how many favor the proposed budget. A company realizes that about 10% of its packages are not being sealed properly. In a case of 24, is it likely that more than 3 are unsealed?

Answers No. More than two outcomes are possible. Yes No. The trials are not independent. No. 500 is more than 10% of 3000 Yes (as long as packages are independent)

Geometric Probability Model for Bernoulli Trials We use a geometric probability model when we want to model how long it will take to achieve the first success in a series of Bernoulli trials.

Geometric Probability Model p = probability of success q = probability of failure = 1-p X = number of trials until the first success 𝑃 𝑋 = 𝑞 𝑥−1 𝑝

Geometric Probability Model p = probability of success q = probability of failure = 1-p X = number of trials until first success 𝑆𝐷 𝑜𝑟 𝜎= 𝑞 𝑝 2 𝐸 𝑋 𝑜𝑟 𝜇= 1 𝑝

Example: What is the probability that if you roll a die repeatedly, the first time you get a 5 is on the 7th roll? 𝑝= 1 6 𝑞= 5 6 𝑃 𝑋=7 = 5 6 7−1 1 6

= 1 – 0.8324 = 0.1676 P(1) = .2 P(≥9) = whatever amount remains Example: Your little brother is opening cereal boxes one at a time trying to find the blue prize, which is in 20% of the boxes. What’s the probability that he finds it in the first box? The second? The third? Etc. P(1) = .2 P(2) = .8(.2) = 0.16 P(3) = .8 2 (.2) = 0.128 P(4) = .8 3 (.2) = 0.1024 P(5) = .8 4 (.2) = 0.082 P(6) = .8 5 (.2) = 0.066 P(7) = .8 6 (.2) = 0.052 P(8) = .8 7 (.2) = 0.042 P(≥9) = whatever amount remains = 1 – 0.8324 = 0.1676 0.8324

X 1 2 3 4 5 6 7 8 ≥9 P(X) Probability Model Example: Your little brother is opening cereal boxes one at a time trying to find the blue prize, which is in 20% of the boxes. What’s the probability that he finds it in the first box? The second? The third? Etc. Probability Model X 1 2 3 4 5 6 7 8 ≥9 P(X) 0.2 0.16 0.128 0.1024 0.082 0.066 0.052 0.042 0.1676

Activity: You are opening boxes of cereal one at a time looking for a prize which is in 20% of boxes. You want to know how many boxes you might have to open in order to find the prize. Describe how you would simulate the search for Tiger using random numbers. Run at least 30 trials. Based on your simulation, estimate the probabilities that you might find your first prize in the first box, the second box, etc. Calculate the actual probability model. Compare the distribution of outcomes in your simulation to the probability model.

Today’s Assignment: Read Chapter 17 Add to HW p.401 #2, 4 Chapter 17 HW Due Friday Unit 4 Test Tuesday, Dec 15 Final review packet by Friday