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 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 in a distribution always add up to 1.

    Circle one:   True   False

  3. Two dice are rolled. Why is each of the 36 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 as one another
    • because each die is fair and the two rolls do not affect each other

  4. Two fair dice are added. What is P(total = 7)?

    • 1/6
    • 1/12
    • 7/36
    • 1/36

  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. Why is the two-dice distribution not the same as counting how often each total came up in 36 actual rolls?

    • because the distribution is worked out beforehand and the counts are what happened
    • because 36 rolls is not enough to see all eleven of the possible totals
    • because the actual rolls would have to be added into the distribution afterwards somehow
    • because the actual rolls use two dice while the distribution uses only one

  8. X takes 1, 2, 3, 4 with chances 0.2, 0.3, 0.4, 0.1. Find P(X at least 3).

    Answer: ______________

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

    Answer: ______________

  10. Dee has the two-dice distribution and is asked for the chance the dice show a double. What should she say?

    • 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 total's distribution cannot answer it, since doubles are spread across several totals
    • 0, because the distribution of the total never mentions doubles anywhere in it at all, ever
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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 · It is the complete description before anything happens: which numbers are possible, and how likely each one is. Counts from an actual experiment are a different thing.
  2. True · The listed values cover every outcome and no outcome produces two values, so the whole of the chance is shared out among them with nothing left over.
  3. because each die is fair and the two rolls do not affect each other · Fairness makes each face equally likely on one die, and independence lets the two multiply, so every pair carries the same 1 / 36. The totals are certainly not equally likely.
  4. 1/6 · Six of the 36 equal pairs total 7, so 6 over 36 = 1/6.
  5. that a value has been left out, or one of the entries is wrong · A complete distribution always totals 1. Falling short means something the variable can do is missing from the list, or a listed probability is not what it should be.
  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. because the distribution is worked out beforehand and the counts are what happened · The distribution says what the game is; the counts say what one run of it did. Thirty-six rolls would give something near the distribution and almost never exactly it.
  8. 0.5 · Add the chances for 3 and 4: 0.4 + 0.1 = 0.5.
  9. 0.2 · The three must total 1, so 0.4 + 2k + k = 1, giving 3k = 0.6 and k = 0.2.
  10. the total's distribution cannot answer it, since doubles are spread across several totals · The distribution of the total forgets which faces came up. Doubles give totals of 2, 4, 6, 8, 10 and 12, and the totals 4, 6, 8 and 10 also contain pairs that are not doubles.
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