Please write neatly and *always show your work*.

- You have a red die and a green die, and will roll both.

- a) There are thirty-six possible outcomes. Show how this number is calculated.

- b) List those thirty-six possibilities.

- c) Explain why each of the thirty-six possibilities should have equal probability.

- d) Let D = the green die result is larger than the red die result. List the elements of D, and then find P(D).

- e) Let E = the green die result is even and the red die result is odd. List the elements of E, then find P(E).

- f) Let F = D
*∩*List the elements of F, then find P(F).

- g) Using what has already been done, find P(D
*∪*E).

- Bea and Zach meet to play three sets of tennis every Saturday. We can think of each of their Sat-urday matches as an experiment which generates an observation. For instance, BZZ would be the observation that Bea won the first set, Zach the second, and Zach the third.

- a) Write down the sample space S for this experiment.

- b) To date, Bea and Zach have played 104 matches (every Saturday for two years), and they have recorded the member of S that resulted each time. Suppose we randomly select an observation from this list of 104, which of course will be a member of S. For this selection, is it reasonable to assign equal probabilities to the members of S? Explain.

- c) Explain how you will assign probabilities to the members of S for the match that Bea and Zach will play next Saturday.

- Three distinct integers are chosen at random from
*{*1*,*2*,*3*, . . . ,*20*}*.

- a) Find the probability that their sum is even.

- b) Find the probability that their product is even.

- A coin is to be tossed as many times as necessary to turn up one head. Thus the elements of the

sample space S are H, TH, TTH, TTTH, etc. To these simple events let’s assign the probabilities 1/2*,* 1/4*,* 1/8*,* 1/16,……

- Is this a discrete sample space? Explain.
- Show that P(S)=1.
- Let A = “at most five tosses to turn up one head.” Find P(A).
- Let B = “the head occurs on the fifth or sixth toss.” Find P(B).
- What (in words) is the event A
*∩*B? Find P(A*∩*B). - What (in words) is the event A
*∪*B? Find P(A*∪*B).

- A committee of four is to be selected from a group of four men and five women.

- a) How many possible committees are there? If the selection of the committee is at random,

- b) What is the probability that there are four five women on the committee?

- c) What is the probability that there are more men than women on the committee?

- How many three-digit numbers can be formed from the digits 0,1,2,3,4,5,6,7 if

- a) each digit can only be used once?

- b) each digit can only be used once and the first digit cannot be zero?

Suppose you are assigned a randomly selected 3-digit number from among those in (b).

- What is the probability that your number is less than 400?
- What is the probability that your number is even?

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