Binomial, Poisson & Normal Distributions


Exercises for students
Solve the following problems on paper and check your answers using the above applet:

Given X ~ Bin(7, 0.2). Calculate P(X = 2), P(2 <= X <= 3), P(X >= 2), P(X > 2).

Given X ~ Po(5). Calculate P(X = 2), P(2 <= X <= 3), P(X >= 2), P(X > 2).

In a lottery there are many tickets sold, in which the chances of winning a prize is 0.1. I bought 25 tickets for this lottery. Calculate the probabilities that
(a) I will not win any prize,
(b) I will win 1 prize,
(c) I will win at most 2 prizes,
(d) I will win at least 3 prizes.

The number of accidents per week on a particular stretch of road follows a Poisson distribution with mean 1.
(a) Find the probability that in a particular week there is exactly 1 accident.
(b) Find the probability that in a particular week there is at least 1 accident.
(c) Find the probability that in a 4 week period there are exactly 4 accidents.

Summary
If X ~ Bin(n, p), then P(X = r) = ⁿC_r p^r q^n-r
If X ~ Po(lambda), then P(X = r) = e^-lambdalambda^r / r!


Examples of Binomial Distributions:

No. of heads obtained from 5 throws of a coin
No. of sixes obtained from 5 throws of a die
No. of lucky draw tickets out of 20 that win prizes
No. of rotten apples out of 20
No. of defective articles out of 20
No. of people out of 20 with a certain blood group
No. of people out of 20 with birthdays in January
No. of people out of 100 who wears glasses


Examples of Poisson Distributions:

No. of cups of coffee sold per day
No. of TV sets sold per week
No. of customers per day
No. of phone calls per day
No. of letters received per week
No. of road accidents per week
No. of injuries in a factory per week
No. of plane crashes per year
No. of flaws in a roll of wallpaper
No. of cases of broken noses seen by a doctor in a year


Examples of Normal Distributions:

Diameter of bolts, metal rods
Mass of cans, jam jars, cereals, potatoes, grapes, coffee, teabags etc
Volume of paint, wine etc
Height of men, women etc
Weight of men, women etc
Length of tiles
Distance walked by a postal worker
Breaking strength of components
Range of round fired from a gun
Time of arrival of train, bus etc
Time to complete assignment, travel to school, repair computers etc
Lifespan of batteries, light bulbs etc





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