# Normal an Poisson approximation to binomial distribution

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In each month, the proportion of "Prize" bonds that win a prize is 1 in 11000. There is a large number of prizes and all bonds are equally likely to win each prize. Show that, for a given month, the probability that a bondholder with 5000 bonds wins at least one prize is 0.365.

For a given month find:

1) The probability taht in a group of 10 bondholders each holding 5000 bonds, four or more win at least one prize,

2) The probability in a group of 100 bondholders each holding 5000 bonds, 40 or more win at least one prize.

Find the expected number of prizes for a bondholder holding 550 bonds for 24 months.

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##### Solution Summary

In each month, the proportion of "Prize" bonds that win a prize is 1 in 11000. There is a large number of prizes and all bonds are equally likely to win each prize. Show that, for a given month, the probability that a bondholder with 5000 bonds wins at least one prize is 0.365.

For a given month find:

1) The probability taht in a group of 10 bondholders each holding 5000 bonds, four or more win at least one prize,

2) The probability in a group of 100 bondholders each holding 5000 bonds, 40 or more win at least one prize.

Find the expected number of prizes for a bondholder holding 550 bonds for 24 months.

##### Solution Preview

Let X be the number of prizes won.

X ~ Bin(5000, 1/11000)

Since n is large, p is very small,

such that np = 5/11

We use Poisson approximation to binomial distribution.

X ~ P(5/11) approx

P( X >= 1) = 1 - P(X=0)

= 1 - e^-5/11

= 0.365 ...

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