Answer to Question #135377 in Statistics and Probability for Kistam Priyanka

Question

Suppose height to the bottom of clouds is a gaussian random variable X for which mean ax = 4000 m, and standard deviation sx = 1000m. A person bets that cloud height tomorrow will fall in the set A={1000m < Xs 3300m) while a second person bets that height will be satisfied by B= (2000m < Xs 4200m). A third person bets they are both correct. Find the probabilities that each person will win the bet.

Answer 1

$X\sim N(a_X, s_X^2 )$

Given $a_X=4000 m, s_X=1000m$

A first person and a second person will be right when cloud height will fall in the set $A\cap B$ ={2000m < X< 3300m)

$$P(2000<X<3300)=$$

$$=P(Z<\dfrac{3300-4000}{1000})-P(Z\leq\dfrac{2000-4000}{1000})=$$

$$=P(Z<-0.7)-P(Z-\leq2)\approx$$

$$\approx0.24196365-0.02275013\approx0.2192$$

If A first person and a second person will be right then and a third person will be right

The probability that each person will win the bet is $0.2192$.