Question

In a city, there are three colors of taxicabs: black (B), grey (G), and silver (S). In this city, 0.4 of taxis are black, 0.35 of taxis are grey, and 0.25 of taxis are silver.

A pedestrian is hit by a taxi but does not see its color. However, there is a witness who says it was a black taxi. The witness is pretty good at identifying the color of a taxi. The probability the witness will get the color right is 0.8, i.e. the probability is 0.8 that the witness will say a taxi is black if it’s black, the probability is 0.8 that the witness will say a taxi is grey if it’s grey, and the probability is 0.8 that the witness will say a taxi is silver if it’s silver. When the witness gets the color wrong, they are equally likely to give each of the wrong colors.

What is the probability that the taxi was black, given the witness says it was black?  Three decimals

Answer 1

B→black taxi

B'→not a black taxi

G→grey

S→silver

P(S)=0.25

P(G)=0.35

Probabolity of black taxi= 0.40

P(B)= 0.40

The probability the witness will get the color right is 0.8

Now

P(right/B)=0.8

Prob(right/G)=0.8

Prob(right/S)=0.8

Que:. What is the probability that the taxi was black, given the witness says it was black?

Solutions:

Now, using bay's theorem

Prob(B/right)=prob(B)*prob(right/B)/[prob(B)*prob(right/B)+prob(G)*prob(right/G)+prob(S)*prob(right/grey)]

=[0.40*0.8]/[0.40*0.80+0.35*0.80+0.25*0.80]= 0.4

Thanks!