A hydrologist records weather–related events over four consecutive days as follows: R…

MATHEMATICS · Multi-part · A Level · Mathematics · Probability · Probability

Problem

A hydrologist records weather–related events over four consecutive days as follows: R

_1

: it rains on Monday, R

_2

: it rains on Tuesday, S: a severe storm occurs on Wednesday, F: the river floods on Thursday. The observed probabilities are:

P(R_1)=0.30

;

P(R_2\mid R_1)=0.60

P(R_2\mid R_1^{c})=0.40

;

P(S\mid R_2)=0.50

P(S\mid R_2^{c})=0.20

;

P(F\mid S)=0.70

P(F\mid S^{c})=0.10

. (i) Develop an expression for

P(F)

entirely in terms of the given conditional probabilities. (ii) Evaluate the numerical value of

P(F)

. (iii) Calculate the probability that the river floods on Thursday even though it did not rain on Monday.