Question

Let $Y$ be the outcome when a single and fair die is rolled. If $E\left(Y\right)=3.5$ and $var\left(Y\right)=\frac{35}{12}$. Evaluate $P(Y\ge 6)$ using Tchebycheff's inequality.

• A. $0.248$
• B. $0.318$
• C. $0.417$
• D. None of these

Answer 1

Answer: (B)

$0.318$

Given ${\sigma }^2=\frac{35}{12}$

$\Rightarrow \mu =E\left(Y\right)=3.5\Rightarrow \mu +a=6\Rightarrow 3.5+a=6\Rightarrow a=2.5$

Using Chebycheff's inequality, we have

$P(Y\ge \mu +a)\le \frac{{\sigma }^2}{{\sigma }^2+a^2}$

$P(Y\ge 6)=P(Y\ge 3.5+2.5)\le \frac{\frac{35}{12}}{\frac{35}{12}+(2.5{)}^2}\le .318$

So, option B is correct.