606 Chapter 21Probability and statistics
More generally, the binomial distribution applies to any experiment whose
outcomes can be treated as two exclusive events. These are often called ‘success’, with
probability p, and ‘failure’, with probabilityq 1 = 111 − 1 p. The probability of obtaining m
successes from ntrials is then
(21.19)
The mean, variance and standard deviation of the distribution are
(21.20)
EXAMPLE 21.6The probability that the outcome of 10 tosses of a coin consists of
mheads (and 101 − 1 mtails) is
with μ 1 = 15 , V 1 = 1 2.5, and σ 1 = 1 1.58. The corresponding binomial distribution is
compared in Table 21.5 with the relative frequenciesf
mobtained from Table 21.2.
Table 21.5
m 0 12345678910
P
m0.001 0.01 0.04 0.12 0.21 0.25 0.21 0.12 0.04 0.01 0.001
f
m0.02 0.00 0.10 0.10 0.18 0.20 0.18 0.12 0.09 0.04 0.00
The mean, variance, and standard deviation of the frequency distribution are
M1= 1 4.98,V(n) 1 = 1 3.8,ands 1 = 1 1.95. The differences between theory and experiment
are due to the small size of the sample (50 sets of 10 tosses).
EXAMPLE 21.7Find the probability of throwing at least 2 ‘sixes’ in 4 throws of a
fair die.
The probability of success in one throw is , so that. The total probability of
at least two successes (two, three or four) is then
P 1 = 1 P
21 + 1 P
31 + 1 P
40 Exercises 12, 13
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10μσ=〈 〉=mnp Vnnp p, ()()= 11 − , = np p()−
P
n
m
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n
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mmnm m nm==
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