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 6 independent trials, each has success probability $\frac{1}{3}$. Let $X$ be the number of successes. Find $P(X=2)$ exactly and the mean of $X$.

Worked explanation

• State the model

  $X\sim\operatorname{Bin}(6,\frac{1}{3})$. Independence and a common success probability are given.

• Apply the probability formula

  $P(X=2)=\binom{6}{2}(\frac{1}{3})^2(\frac{2}{3})^{4}=\frac{80}{243}$.

• Find the mean

  $E(X)=np=2$.

$P(X=2)=\frac{80}{243}$, $E(X)=2$.

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