MAT 003-A and B
INTRODUCTION TO PROBABILITY THEORY AND STATISTICS
Miscellaneous Problems
1. A and B play 12 games of chess of which 6 are won by A, 4 are won by B, and 2 end in a draw. They agree to play a tournament consisting of 3 games.
Find the probability that (a) A wins all 3 games, (b) 2 games end in a draw, (c) A and B win alternately, (d) B wins at least 1 game.
2. A machine produces a total of 12, 000 bolts a day, which are on the average 3% defective. Find the probability that out of 600 bolts chosen at random, 12 will be defective.
3. A box contains 5 red and 4 white marbles. Two marbles are drawn successively from the box without replacement, and it is noted that the second one is white.
What is the probability that the first is also white?
4. Find the probability that n people (n ≤ 365) selected at random will have n different birthdays.
5. Suppose that the random variables X and Y have a joint density function given by
f (x, y) =
(c(2x + y), 2 < x < 6, 0 < y < 5
0, otherwise.
Find
(a) the constant c,
(b) the marginal distribution functions for X and Y , (c) the marginal density functions for X and Y , (d) P (3 < X < 4, Y > 2),
(e) P (X > 3), (f) P (X + Y > 4).