30.8 IMPORTANT DISCRETE DISTRIBUTIONS
1 1
1 1
2 2
2 2
3
3
3
3
44
4 4
5
5 5
(^5678910678910)
0 0
0
00
0
0 0
0. 1 0.^1
0. 1 0. 1
0. 2 0. 2
0. 2 0.^2
0. 3 0.^3
0. 3 0. 3
0. 4 0. 4
0. 4 0.^4
x xx xf(x) f(x)f(x) f(x)n=5,p=0. (^6) n=5,p=0. 167
n= 10,p=0. 6 n= 10,p=0. 167
Figure 30.11 Some typical binomial distributions with various combinations
of parametersnandp.
number of permutations ofnobjects, of whichxare identical and of type 1 and
n−xare identical and of type 2, is given by (30.33) as
n!
x!(n−x)!
≡nCx.
Therefore, the total probability of obtainingxsuccesses fromntrials is
f(x)=Pr(X=x)=nCxpxqn−x=nCxpx(1−p)n−x, (30.94)
which is thebinomial probability distribution formula. When a random variable
Xfollows the binomial distribution forntrials, with a probability of successp,
we writeX∼Bin(n, p). Then the random variableXis often referred to as a
binomialvariate. Some typical binomial distributions are shown in figure 30.11.
If a single six-sided die is rolled five times, what is the probability that a six is thrown
exactly three times?
Here the number of ‘trials’n= 5, and we are interested in the random variable
X= number of sixes thrown.
Since the probability of a ‘success’ isp=^16 , the probability of obtaining exactly three sixes
in five throws is given by (30.94) as
Pr(X=3)=