Statistics Question

 

 

 

Please write neatly and always show your work.

 

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

 

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

 

  1. b) List those thirty-six possibilities.

 

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

 

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

 

  1. 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).

 

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

 

  1. g) Using what has already been done, find P(D E).

 

  1. 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.

 

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

 

  1. 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.

 

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

 

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

 

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

 

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

 

  1. 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,……

 

 

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

 

 

 

 

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

 

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

 

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

 

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

 

 

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

 

  1. a) each digit can only be used once?

 

  1. 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).

 

  1. What is the probability that your number is less than 400?
  2. What is the probability that your number is even?
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