701 words - 3 pages

Probability Discrete Event Simulation:

Tutorial on Probability and Random Variables

1. A card is drawn from an ordinary deck of 52 playing cards. Find the probability that it is (a) an

Ace, (b) a jack of hearts, (c) a three of clubs or a six of diamonds, (d) a heart, (e) any suit except

hearts, (f) a ten of spade, (g) neither a four nor a club

(1/3, 1/52, 1/26, 1/4, 3/4, 4/13, 9/13)

2. A ball is drawn at random from a box containing six red balls, 4 white balls and 5 blue balls. Determine the probability that it is (a) red, (b) white, (c) blue, (d) not red, (e) red or white

(2/5, 4/15, 1/3, 3/5, 2/3)

3. Three balls are drawn successively from the box in the previous problem. Find the probability that

they are drawn in the order red, white and blue if each ball is (a) replaced (b) not replaced

(8/225, 4/91)

A

fair coin is tossed. If the coin turns up heads a marble is chosen from box I and if it turns up tails,

a marble is chosen from box II. Find the probability that a red marble is chosen.

(2/5)

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Discrete Event Simulation:

Tutorial on Probability and Random Variables

8. Suppose in the previous problem, the one who tosses the coin does not reveal whether it has turned

up heads or tails (so that the box from which a marble was chosen is not revealed) but does reveal

that a red marble was chosen. What is the probability that box I was chosen?

(3/4)

9. 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 three

games, (b) two games end in a draw, (c) A and B win alternately, (d) B wins at least one game.

(1/8, 5/72, 5/36, 19/27)

10. 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 1 is

also white?

(3/8)

11. Suppose that a pair of fair dice are tossed and let the random variable X denote the sum of the

points. Obtain the probability distribution of X.

(2 to 12 with probabilities: 1/36,2/36,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36)

12. Find the expectation of the sum of points in tossing pair of fair dice

(7)

13. A continuous random variable X has probability density given by

2e−2x , x > 0;

f (x) =

0,

x≤0.

Find (a) E(X), (b) E(X 2 )

(1/2, 1/2)

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Discrete Event Simulation:

Tutorial on Probability and Random Variables

14. For the previous problem, obtain the variance and standard deviation.

(1/4, 1/2)

15. Find (a) the variance and (b) the standard deviation of the sum obtained in tossing a pair of fair

dice.

(35/6,

35/6)

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