2018 H2 Mathematics Paper 2 Question 6

The Binomial Distribution

Answers

{\frac{5}{9} < p < \frac{2}{3}}

{0.430}

Let

{X}

denote the r.v. of the number of times the bug takes the left fork in the game

X \sim \textrm{B}(8,p)

\begin{align*} & \textrm{P}(\textrm{finishes at } D) \\ & = \textrm{P}(X = 5) \\ & = {8 \choose 5} p^5 (1-p)^{8-5} \\ & = 56 p^5 q^3 \; \blacksquare \end{align*}

\begin{align*} & \textrm{P}(\textrm{finishes at } C) \\ & = \textrm{P}(X = 6) \\ & = {8 \choose 6} p^6 (1-p)^{8-6} \\ & = 28 p^6 q^2 \end{align*}

\begin{align*} & \textrm{P}(\textrm{finishes at } E) \\ & = \textrm{P}(X = 4) \\ & = {8 \choose 4} p^4 (1-p)^{8-4} \\ & = 70 p^4 q^4 \end{align*}

Since the probability that the bug finishes at

{D}

is greater than at any other endpoint,

\begin{gather*} \textrm{P}(\textrm{finishes at } D) > \textrm{P}(\textrm{finishes at } C) \\ 56 p^5 q^3 > 28 p^6 q^2 \\ 2 q > p \\ 2 (1-p) > p \\ p < \frac{2}{3} \end{gather*}

\begin{gather*} \textrm{P}(\textrm{finishes at } D) > \textrm{P}(\textrm{finishes at } E) \\ 56 p^5 q^3 > 70 p^4 q^4 \\ 4 p > 5 q \\ 4 p > 5 (1-p) \\ p > \frac{5}{9} \end{gather*}

Hence the possible range of values of

{p:}

\frac{5}{9} < p < \frac{2}{3} \; \blacksquare

At each fork, the probability of not being swallowed up by a black hole is

{0.9}

\begin{align*} & \textrm{P}(\textrm{not swallowed by black hole}) \\ & = (0.9)^8 \\ &= 0.430 \; \blacksquare \end{align*}