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Minimum and maximum correlation

If X, Y and Z are three random variables such that X and Y have a correlation of 0.9, and Y and Z have correlation of 0.8, what are the minimum and maximum correlation that X and Z can have?

(Thanks to jiantao.)

Solution

The correlation matrix

must be positive semi-definite. A bit of fooling around with that concept will result in the following constraints

+ pxy pyz ┼/i>

For this particular example we have 0.4585 < pxz < 0.9815. It is interesting how small the correlation can be, less than one half, considering how high the other two correlations are. Of course, if one of the two correlations is exactly 1 then this forces the third correlation to be the same as the other.

Airforce One

One hundred people are in line to board Airforce One. There are exactly 100 seats on the plane. Each passenger has a ticket. Each ticket assigns the passenger to a specific seat. The passengers board the aircraft one at a time. GW is the first to board the plane. He cannot read, and does not know which seat is his, so he picks a seat at random and pretends that it is his proper seat.

The remaining passengers board the plane one at a time. If one of them finds their assigned seat empty, they will sit in it. If they find that their seat is already taken, they will pick a seat at random. This continues until everyone has boarded the plane and taken a seat.

What is the probability that the last person to board the plane sits in their proper seat? (Thanks to Wilbur.)

Solution

First of all let me say that the President is now BO, not GW, but then the gag about not being able to read wouldn't work. This problem sounds really complicated, because of all the people who could have sat in the last person s seat before their turn. Start by considering just two people, GW and you. If GW sits in his own seat, which he will do 50% of the time, then you are certain to get your allocated seat. But if he sits in your seat, again with 50% chance, then you are certain to not get the right seat. So apriori result, 50% chance. Now if there are three people, GW either sits in his own seat or in your seat or in the other person s seat. The chances of him sitting in his own seat or your seat are the same, and in the former case you are certain to get your correct seat and in the latter you are certain to not get it. So those two balance out. If he sits in the other person s seat then it all hinges on whether the other person then sits in GW s seat or yours. Both equally likely, end result 50-50 again. You can build on this by induction to get to the simple result that it is 50-50 whether or not you sit in your allocated seat.

Hit-and-run taxi

There was a hit-and-run incident involving a taxi in a city in which 85% of the taxis are green and the remaining 15% are blue. There was a witness to the crime who says that the hit-and-run taxi was blue. Unfortunately this witness is only correct 80% of the time. What is the probability that it was indeed a blue car that hit our victim? (Thanks to orangeman44.)

Solution

A classic probability question that has important consequences for the legal and medical professions.

Suppose that we have 100 such incidents. In 85 of these the taxi will have been green and 15 blue, just based on random selection of taxi colour. In the cases where the taxi was green the witness will mistakenly say that the car is blue 10% of the time, i.e. 17 times. In the 15 blue cases the witness will correctly say blue 80% of the time, i.e. 1i times. So although there were only 15 accidents involving a blue taxi there were 19 reports of a blue taxi being to blame, and most of those (17 out of 19) were in error. These are the so-called false positives one gets in medical tests.

Now, given that we were told it was a blue taxi, what is the probability that it was a blue taxi? That is just 11/19 or 41.4%.

 
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