Question

A contest consists of predicting the results of the win, draw or defeat of $7$ football matches. A sent his entry by predicting at random. The probability that his entry will contain exactly $4$ correct predictions is

• A. $\frac{8}{3^7}$
• B. $\frac{16}{3^7}$
• C. $\frac{280}{3^7}$
• D. $\frac{560}{3^7}$

Answer 1

The correct option is C

$\frac{280}{3^7}$

An explanation for the correct answer:

Step 1: Find the possibility of success and failure.

We have been given that, a contest consists of predicting the results of the win, draw or defeat of $7$ football matches and a sent his entry by predicting at random.

We need to find the probability that his entry will contain exactly $4$ correct predictions.

Let event $A$: his entry contains exactly $4$ correct predictions.

For this event, $p=\frac{1}{3},q=\frac{2}{3}$and $n=7$

Step 2: By using the Binomial distribution formula, the probability that his entry will contain exactly $4$ correct predictions.

$$\begin{gathered} \Rightarrow P\left(A\right)={}^{7}C_4\times {\left(\frac{1}{3}\right)}^4\times {\left(\frac{2}{3}\right)}^{7-4} \\ \Rightarrow P\left(A\right)=\frac{7!}{4!(7-4)!}\times \left(\frac{1}{81}\right)\times {\left(\frac{2}{3}\right)}^3 \\ \Rightarrow P\left(A\right)=35\times \frac{1}{81}\times \frac{8}{27} \\ \Rightarrow P\left(A\right)=\frac{280}{3^4\times 3^3} \\ \Rightarrow P\left(A\right)=\frac{280}{3^7} \end{gathered}$$

Therefore, option (C) is the correct answer.