Discrete Random Variables and Probability Distributions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Every value and the chance of each

Mathematics · Probability · ages 16-17
Name ______________________   Date ____________

  1. What does a discrete probability distribution list?

    • every value the variable can take, with the chance of each
    • the outcomes of the experiment in the order they happened
    • the average value of the random variable over many trials
    • the number of times that each outcome came up during an experiment

  2. The probabilities of all possible values in a discrete probability distribution always add up to 1.

    Circle one:   True   False

  3. Two fair six-sided dice are rolled independently. Why is each of the 36 ordered face pairs equally likely?

    • because there are more pairs than there are totals to spread them over
    • because every total from 2 to 12 has the same chance of coming up
    • because the dice are rolled at the same moment
    • because each die is fair and the two rolls do not affect each other

  4. A coin is flipped three times and X is the number of heads. How many of the 8 equally likely outcomes give X = 2?

    • 2
    • 3
    • 4
    • 6

  5. A distribution's probabilities add to 0.9. What does that tell you?

    • that the random variable is more likely to be small than large
    • that a value has been left out, or one of the entries is wrong
    • that the remaining 0.1 belongs to the largest value in the list
    • that the experiment has to be repeated ten more times to fix it

  6. A distribution lists values 0, 1, 2 and 3 with probabilities 0.1, 0.35, 0.3 and one missing. What is P(X ≥ 2)?

    Answer: ______________

  7. For two fair dice rolled independently, how do the theoretical probabilities differ from the relative frequencies in 36 rolls?

    • The model gives chances; observed frequencies record one run
    • The experiment changes the model probabilities after each roll
    • Every total must occur equally often in any run of 36 rolls
    • The model and observed frequencies must always disagree

  8. For the number of heads X in three flips, P(0) = 1/8, P(1) = 3/8, P(2) = 3/8 and P(3) = 1/8. What is P(X ≥ 2)? Give a decimal.

    Answer: ______________

  9. A distribution lists values 1, 2 and 3 with probabilities 0.4, 2k and k. What is k?

    Answer: ______________

  10. Dee has only a distribution of the total of two six-sided dice. Total 6 has positive probability, but individual face-pair probabilities are unknown. What can she conclude about the chance of a double from this table alone?

    • 6 / 36, reading it straight off the entry for the total of 7 in the finished table
    • 1 / 36, since only a double six would count as a double in this particular game
    • the table cannot separate doubles from non-doubles with the same total
    • 0, because the absence of a separate doubles entry means every matching pair has probability zero
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Every value and the chance of each W1-mt_yKztNkvvq3-s1

  1. every value the variable can take, with the chance of each · A discrete distribution lists every possible value with its probability. Counts recorded during an experiment describe observed results, not the model probabilities.
  2. True · In a discrete distribution, the listed values account for every outcome exactly once. Their probabilities therefore add to 1.
  3. because each die is fair and the two rolls do not affect each other · Fairness gives every face probability 1 / 6. Independence lets you multiply the face probabilities, so every ordered pair has probability 1 / 36. The totals are not equally likely.
  4. 3 · The outcomes with exactly two heads are HHT, HTH and THH, so 3 of the 8.
  5. that a value has been left out, or one of the entries is wrong · A complete distribution always totals 1. A total below 1 means a possible value is missing from the list or a listed probability is incorrect.
  6. 0.55 · The missing entry is 1 - 0.1 - 0.35 - 0.3 = 0.25, so P(X ≥ 2) = 0.3 + 0.25 = 0.55.
  7. The model gives chances; observed frequencies record one run · The model probabilities come from the equally likely pairs. Relative frequencies are the observed counts divided by 36. They may be close to the model or far from it, and exact agreement is possible.
  8. 0.5 · At least 2 means 2 or 3, so 3/8 + 1/8 = 4/8 = 0.5.
  9. 0.2 · The three probabilities must total 1, so 0.4 + 2k + k = 1, giving 3k = 0.6 and k = 0.2.
  10. the table cannot separate doubles from non-doubles with the same total · Total 6 can come from a double, 3 and 3, or a non-double, 2 and 4. Its positive probability does not say how much belongs to either pair. Without face-pair probabilities, the chance of a double is not determined.
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