Problem 4BSC

Poisson Approximation to Binomial Assume that we plan to play the Texas Pick 3 lottery 100 times. For one bet, there is a 1 /1000 probability of winning. If we want to use the Poisson distribution as an approximation to the binomial, are the requirements satisfied? If we use the Poisson distribution to find the probability of 101 wins, we get an extremely small positive number, so is it correct to conclude that the probability of 101 wins is possible, but highly unlikely?

Answer :

Step 1 :

Given,

We have a sample size of n = 100 and for one bet there is a p = 1/1000 probability of winning.

With n = 100 and p = 0.001, the conditions n100 and np 10 (np = 100 0.001 = 0.1) are satisfied. So we can use poisson distribution as an approximation to the binomial.

For 101 wins, the poisson approximation gives small positive probability.

The probability of 101 wins is zero, since it is impossible to have 101 wins in 100 tries.