Construct a probability distribution of the random variable X for the number of false answers in a 4-item true or false test.

Solution:

X = the number of false answers

X = {0,1,2,3,4}

$p=q=\frac 12,n=4$

$P(X=0)=^4C_0(\frac 12)^0(\frac 12)^4=\frac 1{16} \\P(X=1)=^4C_1(\frac 12)^1(\frac 12)^3=\frac 4{16}=\frac 14 \\ P(X=2)=^4C_2(\frac 12)^2(\frac 12)^2=\frac 6{16}=\frac 38 \\ P(X=3)=^4C_3(\frac 12)^3(\frac 12)^1=\frac 4{16}=\frac 14 \\P(X=4)=^4C_4(\frac 12)^4(\frac 12)^0=\frac 1{16}$

Thus, probability distribution of X is:

X P(X)

0 1/16

1 1/4

2 3/8

3 1/4

4 1/16