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.
3.
Find the probability that it takes more than three trips for a guest to catch a lake trout weighing at
least 20 pounds.
4.
What is the expected number of fishing trips that must be taken to catch the first lake trout over 20
pounds? Hint: Use 𝜇 for the geometric distribution and round.
• Warm-up
• Check in homework
A#5.41 pages 217-218 #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
Notes from:
Wednesday
Jan 29
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.
NOTE: 𝑒 ≈ 2.71828
𝜇𝑟 ∙ 𝑒 −𝜇
𝑃 𝑟 =
𝑟!
𝜇 in this formula is the mean
number of successes over time.
Your books uses the Greek letter
lambda, 𝜆.
Requirements
• 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.
Parameters
• The mean is _______.
• The standard deviation is _________________.
Differences
• 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
km2. 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
Example: World War II Bombs
V-1 Buzz Bomb Hits for 576 Regions in South London
Number of Bomb Hits
Probability
Expected # of
Regions
Actual Number of
Regions
0
229
1
211
2
93
3
0.053
30.5
35
4
0.012
6.9
7
5
0.002
1.2
1
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. ____________________
𝜇 =𝑛∙𝑝
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
Press 2nd VARS (to get DISTR)
Select Option B: poissonpdf(
Press ENTER
Enter is the values 𝜇, r
#11 page 219
Pages 219
#s: 10 and 12