The Chemistry Maths Book, Second Edition

21.5 The binomial distribution 605

equal probabilities:. The 4 possible outcomes of 2 tosses are HH,

HT, TH, and TT, each with probability. These outcomes are of three types: 2

heads with probability , 1 head and 1 tail with probability , and 2 tails with

probability. In the same way, the 16 equally probable outcomes of 4 tosses are of 5

types with the following probabilities:

4 heads

3 heads and 1 tail

2 heads and 2 tails

P(2H 1 + 1 2T) 1 = 1 P(HHTT) 1 + 1 P(HTHT) 1 + 1 P(HTTH)

1 head and 3 tails

4 tails

We see that the coefficients of the factor of the probabilities are the numbers

{1, 4, 6, 4, 1} that form the fifth row of the Pascal triangle. They are the binomial

coefficients , for mheads and 41 − 1 mtails. In the case of ntrials, the total number

of possible outcomes is 2

n

(2 per trial) and the probability of mheads andn 1 − 1 mtails is

(21.18)

This collection of probabilities is called a binomial distribution(or Bernoulli

distribution).

Pm n m P

n

m

m

n

(())HT+− = = ×

































1

2

m

4













1

2

4

()

PP() ( )4

1

2

4

T ==TTTT













PPPPP( )()()()()13H T+=+++=×TTTH TTHT THTT HTTT 4

44

1

2













+++=×













PPP()()()TTHH THTH THHT 6

1

2

4

PPPPP( )()()()()31H T+=+++=×HHHT HHTH HTHH THHH 4

44

1

2













PP() ( )4

1

2

1

16

4

H ==HHHH













=

1

4

1

4

1

4

1

2

+=

1

4

1

2

2

1

4

()

=

PP() ()HT==

1

2