Introduction to Health Physics: Fourth Edition - Ruang Baca FMIPA UB

HEALTH PHYSICS INSTRUMENTATION 487
sampling distribution of a population of randomly occurring events is called the
binomial distribution and is given by the expansion of the binomial
(p + q ) n = p n + np n−1 q +
n(n − 1)
p
2!
n−2 q 2 +
n(n − 1)(n − 2)
p
3!
n−3 q 3 +···,
(9.32)
where p is the mean probability of the occurrence of an event, q is the mean probability
of nonoccurrence of the event, p + q = 1, and n is the number of chances
of occurrence. The probability of the occurrence of exactly n events is given by the
first term of the binomial expansion, the probability of occurrence of n − 1 events
is given by the second term, and so on. Using dice as an example, the likelihood of
throwing three ones in three consecutive throws of a die, in which the mean probability
of throwing a one is 1/6, is given by the first term of the expansion, according
to Eq. (9.32), of (1/6 + 5/6) 3 :
3 1 5
+ =
6 6
3 1
+ 3
6
1
6
2
5
+
6
3 × 2
2!
= 1 15 75 125
+ + + = 1
216 216 216 216
1
6
2 5
+
6
3 × 2 × 1
3!
3 5
6
The probabilities of 2 ones, 1 one, and no ones are given by the second, third, and
fourth terms as 15/216, 75/216, and 125/216, respectively. A plot of these probabilities
(Fig. 9-39, curve A), shows the distribution to be very asymmetrical. If we
were to make similar calculations for the probability of throwing 6, 5, 4, 3, 2, 1 or
A
B
Figure 9-39. Probability of throwing 0, 1, 2, and
3 ones in three throws of a die, curve A; and the
probability of throwing 0, 1, 2, 3, 4, 5, and 6 ones
in six throws of a die, curve B.