Paper 2 Section B Practice: Special Discrete Random Variables — Answers

Questions: Special Discrete Random Variables Drills.

Question 1

(a) Each seed has the same germination probability $0.72$, and germination outcomes are independent. The number of trials is fixed at 12 and each trial has two relevant outcomes. Any two valid assumptions earn the marks. [2]

(b)

$$X\sim B(12,0.72).$$

[1]

(c)(i)

$$P(X=9)=\left(\genfrac{}{}{0ex}{}{12}{9}\right)(0.72{)}^9(0.28{)}^3\approx \mathrm{0.251.}$$

[2]

(ii)

$$P(X\ge 9)=\sum _{x=9}^{12}\left(\genfrac{}{}{0ex}{}{12}{x}\right)(0.72{)}^x(0.28{)}^{12-x}\approx \mathrm{0.555.}$$

[2]

(d)

$$E(X)=12(0.72)=8.64,$$ $$\mathrm{Var}(X)=12(0.72)(0.28)=\mathrm{2.4192.}$$

[2]

(e)

$$P(X=9\mid X\ge 9)=\frac{P(X=9)}{P(X\ge 9)}\approx \mathrm{0.453.}$$

[2]

Question 2

(a) Since $E(Y)=20p$,

$$p=\frac{6.4}{20}=\mathrm{0.32.}$$

[2]

(b)

$$P(4\le Y\le 8)=P(Y\le 8)-P(Y\le 3)\approx \mathrm{0.767.}$$

[3]

(c) For one packet, let

$$q=P(Y\ge 9)\approx \mathrm{0.15685.}$$

For $m$ independent packets,

$$P(\text{at least one qualifying packet})=1-(1-q{)}^m.$$

Require

$$1-(1-q{)}^m>0.95,$$

so

$$m>\frac{\ln (0.05)}{\ln (1-q)}\approx \mathrm{17.56.}$$

Since $1-(1-q{)}^{17}\approx 0.9450$ while $1-(1-q{)}^{18}\approx 0.9536$, $m=18$ is the smallest integer. [4]