Problem: Roll a two dice until you get doubles. How long did you have to wait?

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Presentation transcript:

Problem: Roll a two dice until you get doubles. How long did you have to wait?

What was the average waiting time?

What did everyone get?

What is the probability of: 1. Getting doubles on your 1 st roll (wait time of 0)? 2. Getting doubles on your 2 nd roll (wait time of 1)? 3. Getting doubles on your 3 rd roll (wait time of 2)?

Theoretical Distribution: Waiting for doubles….

This is called a Geometric Probability Distribution We are counting the number of failures before our first success on independent trials If the probability of success is p and the probability of failure is q, then the probability of a waiting time of x is given by: Eg) What is the probability of waiting 10 times before you roll doubles?

Try the following: 1. You randomly ask people on the street until you find someone born in December. What is the probability of waiting 6 times for this? 2. The Toronto Maple Leafs have a 1 in 30 probability of winning the Stanley Cup. What is the probability of a waiting time of 50 years? 3. There is a 1 in 7 chance of winning a prize in Roll up the Rim. What is the probability that you will have a waiting time of at least 4 tries before you get a prize?

Expected Waiting Time What is the expected waiting time before you roll doubles on two dice?

A Shortcut for Expected Waiting Time In general, the expected waiting time for a geometric probability is given as: Try: 1. What is the expected waiting time before you roll a six on a dice? 2. What is the expected waiting time before an NHL team wins the championship (there are 30 teams in the league)?

Practice Problems 1. A computer chip has a 5% chance of malfunctioning. a) What is the probability that you will get 20 chips before you find one that malfunctions? b) What is the average waiting time before you find a defective chip? c) What is the probability that your waiting time will be 10 or 11 chips?

Even more practice… 2. You operate a store where 15% of the people who pass by will enter the store. a) On average, how many people go by the store before one person enters? b) What is the probability that your first customer will be the 25 th person to walk by? c) What is the probability that 10 out of 40 people will enter?

Summary Under what conditions can we use the geometric probability distribution? Can we use a geometric probability for the waiting time for drawing cards from a deck until we get a jack? How do you calculate the average waiting time for a geometric probability? Why is a geometric probability called a geometric probability? Pg , 3, 4, 5, 7,18