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(a) Three typed letters and their envelopes are piled on a desk. If someone puts theletters into the envelopes at random (one letter in each), what is theprobabilitythat each letter gets into its own envelope? Call the envelopes A, B, C, and thecorresponding letters a, b, c, and set up the sample space. Note that 鈥渁 in A,b in B, c in A鈥� is one point in the sample space.

(b) What is the probability that at least one letter gets into its own envelope?

Hint: What is the probability that no letter gets into its own envelope?

(c) Let A mean that a got into envelope A, and so on. Find the probability P(A)that a got into A. Find P(B) and P(C). Find the probability P(A + B)that either a or b or both got into their correct envelopes, and the probabilityP(AB) that both got into their correct envelopes. Verify equation (3.6).

Use Bayes鈥� formula (3.8) to repeat these simple problems previously done by usinga reduced sample space.

(a) In a family of two children, what is the probability that both are girls if at

least one is a girl?

(b) What is the probability of all heads in three tosses of a coin if you know that

at least one is a head?

Suppose you have 3 nickels and 4 dimes in your right pocket and 2 nickels and a quarter in your left pocket. You pick a pocket at random and from it select a coin at random. If it is a nickel, what is the probability that it came from your right pocket?

Two dice are thrown. Given the information that the number on the first die iseven, and the number on the second is $\mathbf{<}\mathbf{4}$, set up an appropriate sample space andanswer the following questions.

(a) What are the possible sums and their probabilities?

(b) What is the most probable sum?

(c) What is the probability that the sum is even?

(a) There are 3 red and 5 black balls in one box and 6 red and 4 white balls in another. If you pick a box at random, and then pick a ball from it at random, what is the probability that it is red? Black? White? That it is either red or white?

(b) Suppose the first ball selected is red and is not replaced before a second ball

is drawn. What is the probability that the second ball is red also?

(c) If both balls are red, what is the probability that they both came from the same box?

Consider the set of all permutations of the numbers 1, 2, 3. If you select a permutationat random, what is the probability that the number 2 is in the middle position?In the first position? Do your answers suggest a simple way of answering the same questions for the set of all permutations of the numbers 1 to 7?

(a) Find the probability density function $f\left(x\right)$for the position x of a particle which is executing simple harmonic motion on $(鈭�a,a)$along the x axis. (See Chapter $7$ , Section $2$ , for a discussion of simple harmonic motion.) Hint: The value of x at time t is $x=acos蝇t$. Find the velocity $\frac{dx}{dt}$ ; then the probability of finding the particle in a given $dx$ is proportional to the time it spends there which is inversely proportional to its speed there. Don鈥檛 forget that the total probability of finding the particle somewhere must be $1$.

(b) Sketch the probability density function $f\left(x\right)$found in part (a) and also the cumulative distribution function $f\left(x\right)$ [see equation $(6.4)$].

(c) Find the average and the standard deviation of x in part (a).

(a) There are 10 chairs in a row and 8 people to be seated. In how many ways can this be done?

(b) There are 10 questions on a test and you are to do 8 of them. In how many

Ways can you choose them?

(c) In part (a) what is the probability that the first two chairs in the row are vacant?

(d) In part (b), what is the probability that you omit the first two problems in the

test?

(e) Explain why the answer to parts (a) and (b) are different, but the answers to

(c) and (d) are the same.

A coin is tossed repeatedly; x = number of the toss at which a head first appears.

Five cards are dealt from a shuffled deck. What is the probability that they are all of the same suit? That they are all diamond? That they are all face cards? That the five cards are a sequence in the same suit (for example, 3, 4, 5, 6, 7 of hearts)?