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Another car probability problem

People seem to live probability problems around here, so here's another one that might be somewhat controversial:

Friday, May 24, 2002

If there was no witness, the probability of the taxi being blue would have only been calculated based on the number of cars of each color.  So, the probability of the car being blue would have been 85%.

Now consider what would happen if the police didn't have any knowledge about the number of cars of each color in the city.  The probability of the taxi being blue could only be calculated by the correctness of the witness's testimony.  The probability of the car being blue in this situation would be 80%.

This problem is similar to finding the probability that a statement is true, given that the same statement has been uttered by 2 people who speak the truth 85% and 80% of the time respectively.

The probability that the statement is FALSE is the probability that both people lied (for if even if one of them spoke the truth, the statement would be true!).  This happens with probability (15% * 20%) = 3%.

So, the probability that the taxi was blue is 97%.

Saturday, May 25, 2002

I don't agree with you.
I agree that 97% is the probability that one of the witnesses will speak the truth. But this also includes cases where the two witnesses give different answer.
But here we already know that both of them are giving the same answer, which means either both are lying or both are speaking truth.
So the probability should be
(both speaking truth)/(both speaking truth +both lying)
= (85%*80%)/(85%*80%+15%*20%)= 96%.

vivek gupta
Sunday, May 26, 2002

If one of them is infact speaking the truth, that would make the statement true wouldn't it?  So, based on both responses, we should be able to say that the statement is true 97% of the time.

Remember that the two people are not simply saying the same thing under pressure.  They are both trying to recall the color of the taxi and *happen* to say the same thing.  The problem is to figure out if what they're saying is true.  In this case, even if one of them has reported the color correctly, the statement is true.

I think the probability is 97%. 

Sunday, May 26, 2002

The point is, now we know that they have given the same answer. We have to calculate probability taking into account this knowledge also and due to this knowledge our sample space has reduced.
To show that your method is not correct, lets consider the following case.
Lets say one of the witnesses still speaks truth 85% of the time but the other witness speaks truth only 1% of the time(and rest 99% time he lies). And both of them again said blue.
Now if we use your method then the probability of taxi being blue will be
85%+1%-85%*1% which is > 85%.
Now if we analyse this result, the probability of taxi being blue increased from 85% due to the second witness also giving the answer blue. But because of the fact that the second witness is wrong 99% of the time, one would expect that the probability of taxi beng blue should decrease when the second witness says it is blue.

vivek gupta
Sunday, May 26, 2002

Actually I think it's intuitive for the probability to go *up* when you introduce the 1% guy in the equation.  It would go further up if you introduced another 1% guy in the equation.  Or even a 0.001% guy for that matter.  The point is all these people speak the truth 1% (or 0.001%) of the time as the case may be.  If you think the probability should go down, how do you explain your answer of 96% which is greater than 85% (and 80%)?

Monday, May 27, 2002

My intution is that if the probability of a new witness speaking truth is > 50%(i.e if a person speaks truth more often than he lies), then the probability should increase, otherwise it should decrease. Because a witness who speaks 1% truth and says blue is equivalent to a witness who speaks 99% truth and is saying that the car was green.
Since in the problem the second guy speaks truth more than 50% of the time, that is why the probability increased.
If we take the case with two witnesses having probability 85% and 1% and calculate by my method then it comes
(85%*1%)/(85%*1% + 15%*99%) = 5.4%

vivek gupta
Monday, May 27, 2002

Here's my final blow to this already dead horse.  Your reasoning makes sense if the 2 people had been forced to utter a statement and we were trying to find the probability based on an established trend (of 85% and 80% truth).  In the context of this problem, however, these people had their own opinion. 

You've gotten me confused now!  Anyone else care to share their opinions?

Monday, May 27, 2002

I'm a great believe in simple tables to solve this sort of thing. But there's no way to get fixed-width text on here, so consider this instead:

There are four possibilities:

1. Car is blue, witness is correct. (0.85 x 0.8 = 0.68)
2. Car is blue, witness is wrong. (0.85 x 0.2 = 0.17)
3. Car is green, witness is correct. (0.15 x 0.8 = 0.12)
4. Car is green, witness is wrong (0.15 x 0.2 = 0.03)

Now, we know that the witness said "blue", so that reduces the possibilities to the following:

1. Car is blue, witness is correct. (0.85 x 0.8 = 0.68)
4. Car is green, witness is wrong (0.15 x 0.2 = 0.03)

Thus, the car is blue with probability 0.68 / 0.71, and green with probability 0.03 / 0.71.

So the probability of it being blue is 95.8%

This is another example of Bayesian Probability -- figuring out the probabilty of something (car being blue) GIVEN THAT something else is known to have happened (witness said blue).


Adrian Gilby
Monday, May 27, 2002

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