Statistical Mechanics - Physics at Oregon State University

230 APPENDIX A. SOLUTIONS TO SELECTED PROBLEMS.
Therefore in the limit N → ∞ we have
(I) lim
N→∞ ɛ1 = ∞
(II) lim
N→∞ ɛ1 = 1
(III) lim
N→∞ ɛ1 = 1 − 1
√ 2π
(IV) lim
N→∞ ɛ1 = 0
and only in the last case will the relative error become small. For a 1% error
1 1 + we need e 12 N −R(N) = 0.99 or approximately (using ex ≈ 1 + x ) we need
1 1 = 0.01 which will be true if N > 8.
12 N
Problem 5.
This is a problem which shows how probabilities change when you add information.
This one is called the principle of restricted choice. Bridge players should
be familiar with it!
Label the door you chose with A, the others with B and C. When you make
your first choice, your chance of finding the car is of course 1
3 . Hence there are
three equally probable scenarios.
(1) A = car, B = 0, C = 0 Probability 1
3
(2) A = 0, B = car, C = 0 Probability 1
3
(3) A = 0, B = 0, C = car Probability 1
3
Next, the game show host points at a door. Now there are six scenarios:
(1a) A = car, B = 0, C = 0 Host chooses B Probability 1
6
(1b) A = car, B = 0, C = 0 Host chooses C Probability 1
6
(2a) A = 0, B = car, C = 0 Host chooses B Probability 1
6
(2b) A = 0, B = car, C = 0 Host chooses C Probability 1
6
(3a) A = 0, B = 0, C = car Host chooses B Probability 1
6
(3b) A = 0, B = 0, C = car Host chooses C Probability 1
6
which are all equally probable. But now we introduce the fact that the host
cannot open the door behind which the car is hidden, which eliminates sequences
2a and 3b. In scenario 2 where the car is behind door B, the host has to choose
door C! His choice is restricted! Hence the four possible sequences of events
are: