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Probability Solved Problems for Beginning Statistics

By — McGraw-Hill Professional
Updated on Aug 12, 2011

Review the following concepts if necessary:

Experiment, Outcomes, and Sample Space

  1. An experiment consists of flipping a coin, followed by tossing a die. Give the sample space for this experiment.
  2. Ans.   One of many possible representations of the sample space is S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}.

  3. Give the sample space for observing a patient's Rh blood type.
  4. Ans.  One of many possible representations of the sample space is S = {Rh, Rh+}.

Tree Diagrams and The Counting Rule

  1. Use a tree diagram to illustrate the sample space for the experiment of observing the sex of the children in families consisting of three children.
  2. Ans.   The tree diagram representation for the sex distribution of the three children is shown in Fig. 4-8, where, for example, the branch or outcome mfm represents the outcome that the first born was a male, the second born was a female, and the last born was a male.

    Probability Solved Problems

  3. A sociological study consists of recording the marital status, religion, race, and income of an individual. If marital status is classified into one of four categories, religion into one of three categories, race into one of five categories, and income into one of five categories, how many outcomes are possible for the experiment of recording the information of one of these individuals?
  4. Ans.   Using the counting rule, we see that there are 4 × 3 × 5 × 5 = 300 outcomes possible.

Events, Simple Events, and Compound Events

  1. For the sample space given in Fig. 4-8, give the outcomes associated with the following events and classify each as a simple event or a compound event.
    1. At least one of the children is a girl.
    2. All the children are of the same sex.
    3. None of the children are boys.
    4. All of the children are boys.
  2. Ans.  The event that at least one of the children is a girl means that either one of the three was a girl, or two of the three were girls, or all three were girls. The event that all were of the same sex means that all three were boys or all three were girls. The event that none were boys means that all three were girls. The outcomes for these events are as follows:

    1. mmf, mfm, mff, fmm, fmf, ffm, fff; compound event
    2. mmm, fff; compound event
    3. fff; simple event
    4. mmm; simple event
  3. In the game of Yahtzee, five dice are thrown simultaneously. How many outcomes are there for this experiment? Give the outcomes that correspond to the event that the same number appeared on all five dice.
  4. Ans.   By the counting rule, there are 6 × 6 × 6 × 6 × 6 = 7776 outcomes possible. Six of these 7776 outcomes correspond to the event that the same number appeared on all five dice. These six outcomes are as follows: (1, 1, 1, 1, 1), (2, 2, 2, 2, 2), (3, 3, 3, 3, 3), (4, 4, 4, 4, 4), (5, 5, 5, 5, 5), (6, 6, 6, 6, 6).

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