Assume in a particular city, 80% of taxis are yellow, and 20% are black. A witne
ID: 3154151 • Letter: A
Question
Assume in a particular city, 80% of taxis are yellow, and 20% are black. A witness remembers the color of their taxi correctly 80% of the time (i.e, if the taxi was black they remember it as black 80% of the time, and if the taxi was yellow they remember it as yellow 80% of the time). Let Y be the event the taxi was yellow, and T be the event the witness remembered the color correctly.
(a) Find the probability that a witness remembers the color correctly.
(b) If a witness remembers the color correctly, what is the probability that the taxi was yellow?
(c) What is the probability a taxi was black and the witness did not remember the color correctly?
(d) What is the probability that a witness remembered the color correctly, or the taxi was black (or both)? I.e, the union of the two events.
Explanation / Answer
(a) Find the probability that a witness remembers the color correctly.
a)
Let C = remembers correctly
As given,
P(C) = 0.80 [ANSWER]
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(b) If a witness remembers the color correctly, what is the probability that the taxi was yellow?
Here, the events "remembers correctly" and "color" are independent, as the probability of remembering correctly is independent of color (always 0.8). Hence,
P(Y|C) = P(Y) = 0.80 [ANSWER]
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(c) What is the probability a taxi was black and the witness did not remember the color correctly?
Here, as the events are independent,
P(B n C') = P(B) P(C') = 0.20*(1-0.80) = 0.04 [ANSWER]
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(d) What is the probability that a witness remembered the color correctly, or the taxi was black (or both)? I.e, the union of the two events.
Here,
P(C U B) = P(C) + P(B) - P(C n B)
= P(C) + P(B) - P(C) P(B)
= 0.80 + 0.20 - 0.80*0.20
= 0.84 [ANSWER]
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