Stat 11 Spring 2001 Hw 1 Solutions covering Sections 4.1, 4.2
1. Write down the sample space an experiment consisting of three tosses of a coin. Is it finite or infinite?
S = {HHH,  HHT,  HT H,HT T,T HH, THT, T TH,T TT }

The sample space is finite.

2. A letter is chosen at random from the word FATALITY. (a) What is the probability it is a vowel? (b) The probability that it is a consonant? (c) The probability that it is neither? The sample space in this situation is S = {F, A, T, L, I, Y }.

We have the corresponding probabilities P(F) = 1/8, P(A) = 2/8, P(T) = 2/8, P(L) = 1/8, P(I) = 1/8, P(Y ) = 1/8.
So we have immediately that the probability it is a vowel is P({A, I}) = 3/8.
Similarly we have immediately that the probability it is a consonant is P({F, T, L, Y }) = 5/8.
Of course, the probability that it is neither is 0, since a letter is either a vowel or a consonant.

3. A dorm resident is channel-surfing. “Dawson’s Creek” is on the WB, “Party of Five” is on Fox, “Who Wants to be a Millionaire” on ABC, and the news in on CNN. Assuming the probability that he watches either Dawson or Millionaire is 0.6, the probability that he watches Dawson or the news is 0.5, and the probability that he watches the news or Millionaire is 0.5, what are the individual probabilities of him watching each show? For simplicity use the symbols D, P, M, N to denote the outcomes of watching Dawson, POF, Millionaire, news. (Disclaimer: schedules are purely imaginary).

In this case, we have the sample space S = {D,P,M,N }. We want to find the corresponding probabilities P(D), P(P), P(M), P(N).

We have that ”the probability that he watches either Dawson or Millionaire is 0.6”, which implies immediately that P({D, M}) = P(D) + P(M) = 0.6. Similarly we have P({D, N}) = P(D) + P(N) = 0.5 and P({N, M}) = P(N) + P(M) = 0.5, and from the general properties of probabilities, that P(D) + P(M) + P(N) + P(P) = 1.
Solving this system of simultaneous linear equations gives us P(D) = 0.3, P(M) = 0.3, P(N) = 0.2, P(P) = 0.2.

4. Consider the following experiment. A coin is tossed twice. If both tosses show heads, the experiment will stop. If one head is obtained in the two tosses, the coin will be tossed one more time, and in the case of both tails in the two tosses, the coin will be tossed two more times. Make a tree diagram and list the sample space.

PICT

The corresponding sample space is S = {HH, HTH, HTT, THH, THT, TTHH, TTHT, TTTH, TTTT}.

5. A campus organisation will select one day of the week for an end-of-year picnic. Assume that the weekdays, Monday through Friday, are equally likely, and that each weekend day, Saturday and Sunday, is twice as likely as a weekday to be selected. Use the symbols SU, M, T, W, TH, F, S, to stand for the seven days of the week, Sunday through Monday (a) Assign probabilities to the seven outcomes. (b) Find the probability a weekday will be selected.

Here, the sample space S = {M, T, W, TH, F, S, SU}. Since we know that P(M) = P(T) = P(W) = P(TH) = P(F), call this common value x. Then P(S) = P(SU) = 2x.. Since the sum of all probabilities is 1, we obtain immediately that 9x = 1 and hence that x = 1/9.

6.
(This is an optional question.)
Write down the sample space for an experiment where a coin is tossed till a head is obtained, then stop. Assume the probability of a head in a single toss is 1/3. Making any assumptions you feel necessary, assign probabilities to the sample space. Can you calculate the sum of these probabilities?

The sample space here is S = { T, TH, TTH,. . . } as stated in class. (No credit if all you wrote is this.)
The corresponding probabilities are

              (  )k
P(TT T ...H) =  2   1       k = 0,1,...
   -- --        3   3
  k times
I am here making the assumption that the individual tosses are independent. If you don’t know what this means, look at the textbook.