Traffic along a certain city street is being studied The dai

Traffic along a certain city street is being studied. The daily traffic level is rated as one of low, medium, or high, depending on how many cars traverse the street in a day. The probability that one or more vehicles are involved in an accident during the day is 0.05, 0.20, and 0.90 for the low, medium, or high traffic levels. The probability that the traffic level will be low on any given day is twice the probability that it will be medium, which is twice the probability that it will be high. a) (4) What is the probability that the traffic level is low on a randomly selected day? b) (6) What is the probability that there will be one or more vehicle accidents on a randomly selected day?

Solution

let x be the probability that traffic level will be high.

then by question the probability that the traffic level is medium will be twice the probability that it will high. hence 2x

now the probability that the traffic level is low will be twice the probability that it will medium. hence 2*2x=4x

but total probability is unity.

so x+2x+4x=1 or, 7x=1 or, x=1/7

let L be the event that the traffic level is low on a randomly selected day.

M be the event that the traffic level is medium on a randomly selected day.

H be the event that the traffic level is high on a randomly selected day.

A be the event that one or more vehicles are involved in an accident on a randomly selected day.

so P[L]=4x=4/7

P[M]=2x=2/7   P[H]=x=1/7

given in the question

P[A|L]=P[one or more vehicles are involved in an accident when traffic level is low]=0.05

P[A|M]=P[one or more vehicles are involved in an accident when traffic level is medium]=0.20

P[A|H]=P[one or more vehicles are involved in an accident when traffic level is high]=0.90

a) probability that the traffic level is low on a randomly selected day is

P[L]=4/7=0.5714 [answer]

b) the probability that there will be one or more vehicle accidents on a randomly selected day is

P[A]=P[A|H]*P[H]+P[A|M]*P[M]+P[A|L]*P[L] [by theorem of total probability]

      =0.90*1/7+0.20*2/7+0.05*4/7=0.21429 [answer]