exactly two successes

Practise exactly two successes with an original VectorPaths question, worked explanation and related VCE Methods practice.

3 practice marks

Original practice question

In 8 independent trials, each has success probability $\frac{1}{4}$. Let $X$ be the number of successes. Find $P(X=2)$ exactly and the mean of $X$.

Worked explanation

• State the model

  $X\sim\operatorname{Bin}(8,\frac{1}{4})$. Independence and a common success probability are given.

• Apply the probability formula

  $P(X=2)=\binom{8}{2}(\frac{1}{4})^2(\frac{3}{4})^{6}=\frac{5103}{16384}$.

• Find the mean

  $E(X)=np=2$.

$P(X=2)=\frac{5103}{16384}$, $E(X)=2$.

Original VectorPaths practice; not an official VCAA question or marking rubric.