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 is $0.8365$.

Step 1. Given Information.

Twelve percent of the population is left handed. 

So, P (left handed) $=0.12$

Step 2. Assumption

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

Step 3. Approximate binomial distribution with normal distribution.

P (left handed) $=0.12$

$\mu =200\times 0.12 = 24$

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

Where, $n=200$ and $p=0.12$

$$\begin{gathered} \sigma =\sqrt{(200\times 0.12)(1-0.12)} \\ \sigma =\sqrt{24\times 0.88} \\ \sigma =\sqrt{21.12} \\ \sigma \approx 4.60 \end{gathered}$$

$$\begin{gathered} P(X\ge 19.5)=P\left(Z\ge \frac{19.5-24}{4.60}\right) \\ =P\left(Z\ge -0.98\right) \\ =1-\left\{1-\phi \left(0.98\right)\right\} \\ =0.8365 \end{gathered}$$