Robert Bartoszynski - Probability and Statistical Inference

An approximation to is given by the so‐called Stirling's formula.

Theorem 3.4.3 (Stirling's Formula) We have

(3.29)

where the sign means that the ratio of the two sides tends to 1 as .

We shall not give the proof here, but interested readers can find it in more advanced texts, for example, in Chow and Teicher (1997).

A group of boys and girls is divided at random into two equal parts. What is the probability and its approximation for large , that boys and girls are divided evenly between the two groups?

Clearly, the number of ways a group can be divided into two equal parts is . The number of ways boys can be divided evenly is and the same holds for girls. Thus,

(3.30)

which, based on ( 3.29), can be approximated by

For example, in the case of 16 boys and 16 girls , the approximate chances of dividing both sexes evenly are about The exact value is 0.2756.

1 3.4.1 Show that if then(i) Use the definition of binomial coefficients as ratios of the factorials. (ii) Use directly the interpretation of the binomial coefficients as the number of subsets of a given size. (iii) How many ways can one choose an ‐element subset from a ‐element subset from a ‐element subset from a ‐element subset from a element set? (where ).

2 3.4.2 Find the coefficient of the term in the expansion of .

3 3.4.3 Use the argument analogous to that in Theorem 3.3.2to show that if , and , then

4 3.4.4 Use Stirling's formula to approximate the number of ways: (i) A set of size can be partitioned into two equal parts. (ii) A set of size can be partitioned into three equal parts.

5 3.4.5 Approximate the probability that every state will be represented in the committee in Problem 3.3.1(i).

1 1The integer part of , [], is the largest integer not exceeding ( etc.).

2 2This property, called exchangeability of events, will be discussed in more detail in Section 4.6.

3 3The discarded cards are not mixed with the deck. Assume that the player receives the replacement of the discarded cards from the unused remainder of the deck.

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