Answer to Question #208974 in Statistics and Probability for mimi

Question

Assume that a weather forecast for tomorrow states that it rains in Johannesburg with probability

0.2, and it rains in Pretoria with probability 0.3. Assume further that we are told that if it does rain

in Pretoria, then it rains in Johannesburg with probability 0.8. Calculate the following:

(a) The probability that it rains in both Pretoria and Johannesburg. (4)

(b) The probability that it rains in at least one of the cities. (4)

(c) The probability that it does not rain in either city. (2)

(d) The probability that it rains in Pretoria, if it rains in Johannesburg

Answer 1

Solution:

$P(J)=0.2,P(P)=0.3,P(J|P)=0.8$

(a):

$P(P\cap J)=P(P)\times P(J)=0.3\times 0.2=0.06$

(b):

Required probability$=P(P)P(J')+P(J)P(P')+P(P)P(J)$

$=0.3\times 0.8+0.2\times0.7+0.2\times0.3 \\=0.24+0.14+0.06 \\=0.44$

(c):

Required probability$=1-0.44=0.56$

(d):

$P(P|J)=\dfrac{P(P\cap J)}{P(J)}=\dfrac{0.06}{0.2}=0.3$