Question

The weight of students ina college has a mean of 40kgs and standard dev of 5 kg.In a random sample of 5 students what is the probability that
1) all have weight>50kg
2) 3 will have weight>50kg

Answer 1

Let $X$ be the weight of a student, and random variable $Z$ has standard normal distribution $Z$ ~$N(0,1)$ .

We assume that $X$ has normal distribution, then $X$ ~$N(40,5^2)$, and

$\frac{X-40}{5}$ ~$Z$.

The probability that student weight $>50$ is

$p=P(X>50)=P(\frac{X-40}{5}>\frac{50-40}{5})=\\ P(Z>2)=1-P(Z\leq2)=\\1-0.9772=0.0228.$

Let $k$ be the number of students weighing $>50$, then $k$ has a binomial distribution

with the probability of success $p=0.0228$ and parameter $n=5$.

a)

$p(k=5)=C_n^kp^k(1-p)^{n-k}=\\ C_5^5*0.0228^5*0.9772^0=0.0228^5\approx 6*10^{-9}.$

b)

$p(k=3)=C_n^kp^k(1-p)^{n-k}=\\ C_5^3*0.0228^3*0.9772^2=\\ \frac{5*4*3}{1*2*3}*0.0228^3*0.9772^2\approx 10^{-4}.$

Answer: a) $6*10^{-9}$ , b) $10^{-4}.$