Q.5.28 - A First Course in Probability

Question

Twelve percent of the population is left handed. Approximate the probability that there are at least $20$ lefthanders in a school of $200$ students. State your assumptions.

Step-by-Step Solution

Answer

The probability that there are at least $20$ lefthanders in a school of $200$ students is $0.8365$. 

Step 1. Given Information.

Here it is given that $12\%$ of the population is left handed.

$P(left handed)=0.12$

Step 2. Basic Assumption

Assuming that except $12\%$ of the population all others are right handed.

Step 3. Approximate binomial distribution with normal distribution.

$$\begin{gathered} P(left handed)=0.12 \\ \mu =200\times 0.12=24 \end{gathered}$$

$\sigma =\sqrt{\left(np\right)\left(1-p\right)}$

Here, $n=200$ and $p=0.12$

$$\begin{gathered} \sigma =\sqrt{\left(200\times 0.12\right)\left(1-0.12\right)} \\ =\sqrt{24\times 0.88} \\ =\sqrt{21.12} \\ =4.6 \end{gathered}$$

$$\begin{gathered} P(X\ge 19.5)=P\left(Z\ge \frac{19.5-24}{4.6}\right) \\ =P(Z\ge -0.98) \\ =1-\left\{1-\phi \left(0.98\right)\right\} \\ =0.8365 \end{gathered}$$

Therefore, the probability that there are at least $20$ lefthanders in a school of $200$ students is $0.8365$.