**Question 1**:

In a school examination, 2 students out of
5 students failed Chemistry.

**(a)**

**If 6 students are chosen at random, find the probability that not more than 2 students failed Chemistry.**

**(b)**If there are 200 Form 4 students in that school, find the mean and standard deviation of the number of students who failed Chemistry.

*Solution:***(a)**

$$\begin{array}{l}X-\text{NumberofstuedentswhofailedChemistry}\text{.}\\ X~B\left(n,p\right)\\ X~B\left(6,\text{}\frac{2}{5}\right)\\ \\ P(X=r)=c{}_{r}.{p}^{r}.{q}^{n-r}\\ P(X\le 2)\\ =P\left(X=0\right)+P\left(X=1\right)+P\left(X=2\right)\\ =C{}_{0}{\left(\frac{2}{5}\right)}^{0}{\left(\frac{3}{5}\right)}^{6}+C{}_{1}{\left(\frac{2}{5}\right)}^{1}{\left(\frac{3}{5}\right)}^{5}+C{}_{2}{\left(\frac{2}{5}\right)}^{2}{\left(\frac{3}{5}\right)}^{4}\\ =0.0467+0.1866+0.3110\\ =0.5443\end{array}$$

**(b)**

$$\begin{array}{l}X~B\left(n,p\right)\\ X~B\left(200,\text{}\frac{2}{5}\right)\\ \text{Meanof}X\\ =np=200\times \frac{2}{5}=80\\ \\ \text{Standarddeviationof}X\\ =\sqrt{npq}\\ =\sqrt{200\times \frac{2}{5}\times \frac{3}{5}}\\ =\sqrt{48}\\ =6.93\end{array}$$

**Question 2**:

5% of the supply of mangoes received by a supermarket are rotten.

**(a)**If a sample of 12 mangoes is chosen at random, find the probability that at least two mangoes are rotten.

**(b)**

**Find the minimum number of mangoes that have to be chosen so that the probability of obtaining at least one rotten mango is greater than 0.85.**

*Solution:***(a)**

*X*~

*B*(12, 0.05)

1 –

*P*(*X ≤*1)
= 1 – [

*P*(*X =*0) +*P*(*X*=*1)]*
= 1 – [ $C{}_{0}$
(0.05)

^{0}(0.95)^{12}+ $C{}_{1}$ (0.05)^{1}(0.95)^{11}]
= 1 – 0.8816

**= 0.1184**

**(b)**

*P*(

*X ≥*1) > 0.85

1 –

*P*(*X =*0) > 0.85*P*(

*X =*0) < 0.15

$C{}_{0}$
(0.05)

^{0}(0.95)^{n}^{ }< 0.15*nlg*0.95 <

*lg*0.15

*n*> 36.98

**n =****37**

Therefore, the minimum number of mangoes
that have to be chosen so that the probability of obtaining at least one rotten
mango is greater than 0.85 is

**37**.**Question 3**:

The result of a study shows that 20% of
students failed the Form 5 examination in a school.

If 8 students from the school are chosen at random, calculate the
probability that

**(a)**exactly 2 of them who failed,

**(b)**less than 3 of them who failed.

*Solution:***(a)**

*p*= 20% = 0.2,

*q*= 1 – 0.2 = 0.8

*X*~

*B*(8, 0.2)

*P*(

*X =*2)

= $C{}_{2}$ (0.2)

^{2}(0.8)^{6}
=

**0.2936****(b)**

*P*(

*X <*3)

=

*P*(*X =*0) +*P*(*X*=*1) +**P*(*X*=*2)*
= $C{}_{0}$
(0.2)

^{0}(0.8)^{8 }+ $C{}_{1}$ (0.2)^{1}(0.8)^{7}+ $C{}_{2}$ (0.2)^{2}(0.8)^{6}
= 0.16777 + 0.33554 + 0.29360

=

**0.79691**