A car crash and escape happened in a city , There are only two colors in the city , The scale is blue 15%  green 85%, A witness saw it at the scene when it happened , He pointed to the blue car
But according to the experts' Analysis on the spot , At that time, the possibility that the eyewitness could see the correct color of the car was 80%
that , What is the probability that the car causing the accident is a blue car （）
A.15%  B.41%  C.80%   D.83%

Baidu found that there is no agreement on the answer on the Internet , There are the following mistakes ：

1. The proportion of two color cars in the city 15%,85% It's confusing conditions , The result should be 80%.

Without considering witnesses , The quantity can represent the probability of occurrence , Then the possibility of causing the accident is 15% blue ,85% green .

This condition has an effect on the actual results , It's not confusing conditions . The two together determine the final result .

2. The witness's point is to confuse the condition , The result should be 15%.

If the specific gravity of the car is not taken into account 15% and 85%, Only according to the witness , The possibility that the accident vehicle will be blue is 80%, This condition affects the results , It's not confusing conditions . The two together determine the final result .

3. There are two possibilities of actually being blue , See the blue as the possibility 80%, Look at the green and see the mistake 20%×15%, Add up 83%

What's the probability that the car is blue 15%, instead of 100%, What's the possibility of seeing it as blue 80%×15%. The witness said it was blue, just the actual result of the car , It's not probability , According to this result, the actual situation can be divided into two categories : It's blue. Look right , It's blue to see the mistake . So there is no case of looking at the wrong thing as green .

blue

green

The actual color of the vehicle without witnesses

15%

85%

The witness saw the right color

80%

80%

The eyewitness saw the wrong color

20%

20%

Witnesses saw the color of the car

Blue looks right , Green look at mistakes

15%×80%+85%×20%=29%

Green is right , See the mistake in blue

85%×80%+15%×20%=71%

In this question , It's based on the possibility that the witness saw the blue car and the witness saw it correctly 80% In the case of , Calculate the probability of the actual color of the vehicle according to the proportion of the number , Is a conditional probability event .

conditional probability ： set up A,B  It's two events , And A It's not impossible , It is called an event A Under the conditions of occurrence , event B Conditional probability of occurrence  .

set up A={ Witnesses saw the car was blue },B={ The actual color of the car is blue }

P(A)=80%×15%+20%×85%=29%

Namely ： The car is blue (15%)× Witnesses see it right (80%)+ The car is green (85%)× The witness was wrong (20%)

P(AB)=80%×15%=12%

Namely ： The car is blue (15%)× Witnesses see it right (80%)

P(B|A)=P(A)/P(AB)=12%/29%≈41%

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