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(ielm151)[2010](s)hw~cwyipac^_56546.pdf
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IELM 151 Engineering Probability and Statistics Jiheng Zhang Assigned March 4 (Fri.), 2011 Due March 15 (Tue.), 2011
Homework #2
1.
An experiment consists of tossing a coin three times. What is the sample space of this experiment? Which event corresponds to the ex-periment resulting in more heads than tails?

2.
Let E, F, G be three events. Find expressions for the events that of E, F, G

(a)
only E occurs;

(b)
both E and G but not F occurs;

(c)
at least one of the events occurs;

(d)
at least two of the events occur;

(e)
all three occur;

(f)
none of the events occurs;

(g)
at most one of them occurs;

(h)
at most two of them occur;

(i)
exactly two of them occur;

(j)
at most three of them occur.



3.
If P (A)=



1 3
and P (Bc)=
1 4
, can A and B be disjoint? Explain.

4. A red die, a blue die, and a yellow die (all six-sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die which is less than that appearing on the red die. (That is , if B(R)[Y ] is the number appearing on the blue (red) [yellow] die, then we are interested in P (B<Y <R).)
(a)
What is the probability that no two of the dice land on the same number?

(b)
Given that no two of the dice land on the same number, what is the conditional probability that B<Y <R?

(c)
What is P (B<Y <R)?

(d)
If we regard the outcome of the experiment as the vector B, R, Y , how many outcomes are there in the sample space?

(e)
Without using the answer to (c), determine the number of out-comes that result in B<Y <R.

(f)
Use the results of parts (d) and (e) to verify your answer to part (c).


1

5. Judge each of the following propositions by true or false. (Assume that any conditioning event has positive probability.)
(a)
If P (B) = 1, then P (A|B)= P (A) for any A.

(b)
If A . B, then P (B|A)=1 and P (A|B)= P (A)/P (B).

(c)
If A and B are mutually exclusive, then


P (A)P (A|A B)=
P (A)+ P (B)
.

(d) P (A B C)= P (A|B C)P (B|C)P (C).
6.
Tom has 7 di.erent books to put on the shelf. 3 math books, 2 physics books, 2 history books. Suppose he wants to arrange the books by subjects on the shelf. How many di.erent arrangements are possible?

7.
A closet contains n pairs of shoes. If 2r shoes are chosen at random (r<n), what is the probability that there will be no matching pair in the sample?


2