 # 8.3.1 Probability Distribution, Short Questions (Question 1 & 2)

Question 1:
Diagram below shows the graph of a binomial distribution of X.

(a) the value of h,
(b) P (X ≥ 3)

Solution:
(a)
P (X = 0) + P (X = 1) + P (X = 2) + P (X = 3) + P (X = 4) = 1
$\begin{array}{l}\frac{1}{16}+\frac{1}{4}+h+\frac{1}{4}+\frac{1}{16}=1\\ h=1-\frac{5}{8}\\ h=\frac{3}{8}\end{array}$

(b)
P (X ≥ 3) = P (X = 3) + P (X = 4)
$P\left(X\ge 3\right)=\frac{1}{4}+\frac{1}{16}=\frac{5}{16}$

Question 2:
The random variable X represents a binomial distribution with 10 trails and the probability of success is ¼.
(a) the standard deviation of the distribution,
(b) the probability that at least one trial is success.

Solution:
(a)
n = 10, p = ¼
$\begin{array}{l}\text{Standard deviation}=\sqrt{npq}\\ \text{}=\sqrt{10×\frac{1}{4}×\frac{3}{4}}\\ \text{}=1.875\end{array}$

(b)
$\begin{array}{l}P\left(X=r\right)=C{}_{r}{\left(\frac{1}{4}\right)}^{r}{\left(\frac{3}{4}\right)}^{10-r}\\ P\left(X\ge 1\right)\\ =1-P\left(X<1\right)\\ =1-P\left(X=0\right)\\ =1-C{}_{0}{\left(\frac{1}{4}\right)}^{0}{\left(\frac{3}{4}\right)}^{10}\\ =0.9437\end{array}$