Discrete Random Variables: Question 9

Worked solution

Step 1: Identify n and p

$X\sim B(20,0.35)$, so $n=20$, $p=0.35$, and $q=1-p=0.65$.

Step 2: Apply the variance formula

For a binomial distribution, $\mathrm{Var}(X)=np(1-p)=npq$

Substituting the values: $\mathrm{Var}(X)=20\times0.35\times0.65$

Compute step by step: $0.35\times0.65=0.2275$ $20\times0.2275=4.55$

So $\mathrm{Var}(X)=\boxed{4.55}$.

Why the other options are wrong

• B ($7$): this is $E(X)=np=20\times0.35=7$, the mean, not the variance. The factor of $(1-p)$ has been left out entirely.
• C ($2.45$): this comes from using $p$ twice, i.e. $np^2=20\times(0.35)^2=2.45$, instead of $np(1-p)$.
• D ($8.45$): this comes from using $(1-p)$ twice, i.e. $n(1-p)^2=20\times(0.65)^2=8.45$, instead of $np(1-p)$.

Final answer

• $\mathrm{Var}(X)=\boxed{4.55}$, option A.