At Fontaine Lake Camp on Lake Athabasca in northern Canada, history shows that about 30% of the guests catch lake trout over 20 pounds on a 4-day fishing trip (Source: Athabasca Fishing Lodges, Saskotoon, Canada.) Let n be a random variable that represents the first trip to Fontaine Lake Camp on which a guest catches a lake trout over 20 pounds. 1.Write out a formula for the probability distribution of the random variable n. 2.Find the probability that a guest catches a lake trout weighing at least 20 pounds for the first time on trip number 3.

Notes from: Wednesday Jan 29 Warm-up Check in homework A#5.41 pages #s 5, 6, and 8 Notes – Section 5.4 – Part 2 Poisson Probability Distribution Homework due Thursday 1/30: A#5.42 page 219 # 10 and 12

After this section, you will be able to: 1. Use the Poisson distribution to compute the probability of the occurrence of events spread out over time or space; 2. Use the Poisson distribution to approximate the binomial distribution when the number of trials is large and the probability of success is small.

Simeon Denis Poisson (1781 – 1840) French mathematician Studied probabilities of rare events that occur infrequently in space, time, volume, etc. Poisson distribution applies to such things as: Arrivals of people in line Planes arriving at an airport Cars pulling into a gas station Diners arriving at a restaurant Internet users logging onto a Web site

The Poisson Distribution is a discrete probability distribution that applies to occurrences of some event ________________________________________. The random variable, _____, is the ____________ of occurrences of the event in an ________________. The _______________ can be time, distance, area, volume, or some similar unit.

The random variable, _____, is the number of occurrences of an event over __________ ____________. The occurrences must be __________________. The occurrences must be __________________ of each other. The occurrences must be _______________ _______________ over the interval being used.

The mean is _______. The standard deviation is _________________.

The binomial distribution is affected by the ______________________ and the ____________________, whereas the Poisson distribution is affected only by the _______________. In a binomial distribution, the possible values of the _________________ are 0, 1, …, n, but a Poisson distribution has possible r values of 0, 1, 2, …. with no _________________.

Example: World War II Bombs In analyzing hits by V-1 buzz bombs in World War II, South London was subdivided into 576 regions, each with an area of 0.25 km 2. A total of 535 bombs hit the combined area of 576 regions. If a region is randomly selected, find the probability that it was hit exactly twice.

Broken down into 576 regions 535 Bomb Strikes in this area

Example: World War II Bombs

V-1 Buzz Bomb Hits for 576 Regions in South London Number of Bomb HitsProbability Expected # of Regions Actual Number of Regions

Poisson as an Approximation to Binomial The Poisson distribution is sometimes used to approximate the binomial distribution when n is __________________ and p is _____________. The following two conditions must be satisfied: 1.____________________ 2.____________________

Example: Roulette Allyn bets on the number 7 for each of 200 spins of a roulette wheel. Because P(7) = 1/38, he expects to win about 5 times. Find the probability that he wins exactly 8 times.

TI-83 PLUS instructions

#11 page 219

Pages 219 #s: 10 and 12