Question

The maximum weight that an elevator in an apartment complex can accommodate is 800kg. The average adult weight be about 70 kgs with a variance of 200. What is the probability that the lift safely reaches the ground when there are 10 adults in the lift?

Answer 1

According to the task, the mean $\mu = 70$ kg, variance $\sigma^2 =200$ kg, $\sigma = \sqrt{200} \approx 14.14$ kg.

Let's assume that the mass of 1 person is distributed normally. Then we need to estimate mass of 10 people. We know that sum of normal distributions is normal distribution. In our case the mass of 10 people will be distributed according to normal distribution with parameters $\mu_{tot} = 10*\mu = 700$ kg and $\sigma^2_{tot} = 10*\sigma^2 = 2000$ kg, $\sigma_{tot}=\sqrt{2000}=44.72$ kg. To find the probability that the lift safely reaches the ground when there are 10 people in the lift we need to take CDF(cumulative distribution function) for normal distribution $N(\mu_{tot}, \sigma^2_{tot})$. CDF shows the probability of random variable to be less then value x, $P(X<x).$

$P(X<800) = \frac{1}{2}[1+erf(\frac{800-\mu_{tot}}{\sqrt{2}\sigma_{tot}})]=\frac{1}{2}[1+erf(\frac{800-700}{63.243})]=0.987$