Specimen 2017 H2 Mathematics Paper 2 Question 7

The Binomial Distribution

Answers

(i)

The probability of a randomly chosen sweet is yellow is the same for each sweet.
Whether a randomly chosen sweet is yellow is independent of any other sweet

${0.886.}$

(ii)

{0.546.}

(iii)

{\frac{2}{7} < p < \frac{3}{7}.}

Full solutions

(i)

The probability of a randomly chosen sweet is yellow is the same for each sweet.
Whether a randomly chosen sweet is yellow is independent of any other sweet

Let ${X}$ denote the r.v. of the number of yellow sweets ini a packet of 6.

X \sim \textrm{B}\left(6, \frac{1}{10}\right)

\begin{align*} \mathrm{P}(X \leq 1) &= 0.88574\\ &= 0.886 \; \blacksquare \end{align*}

(ii)

Let

{Y}

denote the r.v. of the number of packets that contain no more than one yellow sweet out of 90.

X \sim \textrm{B}\left(90, 0.88574\right)

\begin{align*} \mathrm{P}(X \geq 80) &= 1 - \mathrm{P}(X\leq 79) \\ &= 0.546 \; \blacksquare \end{align*}

(iii)

Let

{R}

denote the r.v. of the number of red sweets in a packet of 6.

R \sim \textrm{B}(6, p)

Since the modal number is

{2,}

\begin{align*} \textrm{P} ( R=2 ) &> \textrm{P} (R=1) \\ {6 \choose 2} p^2 (1-p)^4 &> {6 \choose 1} p (1-p)^5 \\ 15 p &> 6 (1-p) \\ p &> \frac{2}{7} \end{align*}

\begin{align*} \textrm{P} ( R=2 ) &> \textrm{P} (R=3) \\ {6 \choose 2} p^2 (1-p)^4 &> {6 \choose 3} p^3 (1-p)^3 \\ 15 (1-p) &> 20 p \\ p &< \frac{3}{7} \end{align*}

Range of values that

{p}

can take:

\frac{2}{7} < p < \frac{3}{7} \; \blacksquare