http://www.ck12.org Chapter 4. Discrete Probability Distribution
λ=The mean number of events during the time, distance, volume or area.
e=the base of the natural logarithm= 2. 718281828 ...
Example:
A lake, popular among boat fishermen, has an average catch of three fish every two hours during the month of
October.
- What is the probability distribution forx, the number of fish that you will catch, in 7 hours?
- What is the probability that you will catch 0, 3 ,or 10 fish in seven hours of fishing?
- What is the probability that you will catch 4 or more fish in 7 hours?
Solution:
- The mean value number is 3 fish in 2 hours or 1.5 fish/1 hour. This means, over seven hours, the mean number
events will beλ= 1 .5 fish/hour * 7 hours = 10.5 fish. Thus,
p(x) =λ
xe−λ
x! =( 10. 5 )xe−^10.^5
x!- To calculate the probabilities that you will catch 0, 3 ,or 10:
p( 0 ) =
( 10. 5 )^0 e−^10.^5
0!≈ 0. 000027 ≈0%
This says that it is almost guaranteed that you will catch fish in 7 hours.
p( 3 ) =
( 10. 5 )^3 e−^10.^5
3!≈ 0. 0052 ≈ 0 .5%
p( 10 ) =( 10. 5 )^10 e−^10.^5
10!≈ 0. 1212 ≈12%
The curve in the figure withλ=10 is very similar graph of the function for this problem.
- The probability that you will catch 4 or more fish in 7 hours is,