Discrete Random Variables: Question 8

Worked solution

Part (a): Identifying the binomial model

Each of the $n=12$ questions is answered independently, and on each one the student guesses correctly with probability $p=\dfrac{1}{5}=0.2$ (one correct answer out of $5$ options), or incorrectly with probability $0.8$.

This fits the binomial model because:

• There is a fixed number of trials, $n=12$ questions.
• Each trial (question) has exactly two outcomes, correct or incorrect.
• The probability of a correct guess is constant, $p=0.2$, for every question.
• The questions are independent, since one guess does not affect any other.

So $X\sim B(12,0.2)$.

Part (b): Finding P(X = 4)

Using $P(X=r)=\binom{n}{r}p^r(1-p)^{n-r}$ with $n=12$, $p=0.2$, $r=4$: $P(X=4)=\binom{12}{4}(0.2)^4(0.8)^{8}$

Compute each piece: $\binom{12}{4}=495, \qquad (0.2)^4=0.0016, \qquad (0.8)^{8}\approx0.16777$

Multiplying: $P(X=4)=495\times0.0016\times0.16777\approx0.1329$

So $P(X=4)\approx\boxed{0.133}$ (to 3 significant figures).

Check by an alternative grouping: $495\times0.0016=0.792$, and $0.792\times0.16777216\approx0.13288$, which rounds to the same $0.133$.

Part (c): Finding E(X) and Var(X)

For a binomial distribution, $E(X)=np$: $E(X)=12\times0.2=2.4$

And $\mathrm{Var}(X)=np(1-p)$: $\mathrm{Var}(X)=12\times0.2\times0.8=1.92$

Check: $12\times0.2=2.4$ and $2.4\times0.8=1.92$, confirming both values.

Final answers

• (a) $n=12$, $p=0.2$, so $X\sim\boxed{B(12,0.2)}$.
• (b) $P(X=4)\approx\boxed{0.133}$
• (c) $E(X)=\boxed{2.4}$, $\mathrm{Var}(X)=\boxed{1.92}$