AP Statistics · Free-response skill

Probability from a two-way table

The move. From a two-way table, compute and compare conditional probabilities, find an expected value and a binomial probability, and say what each means in context.

How to do it

  1. Name the group first. "Given a casual rider" means the denominator is the casual riders' total, not the grand total.
  2. To test independence, put a conditional probability next to the matching overall probability, with both numbers, and say whether knowing one event changes the chance of the other.
  3. For an expected value, list each value with its probability, multiply and add, then say it is the long-run average per trial, with the unit.
  4. For a binomial probability, name n, p and the success, say why the trials are independent, then show the formula or the calculator command with its inputs before the answer.

Practice prompt Try it

A fictional bike-share company in Riverton looked at 500 recent trips. Each trip was taken either by a member or by a casual rider, and each trip was either short (30 minutes or less) or long (more than 30 minutes). The company charges casual riders 3 dollars for a short trip and 7 dollars for a long trip.

500 bike-share trips by rider type and trip length

Rider typeShort tripLong tripTotal
Member26040300
Casual rider12080200
Total380120500

Use the table to answer each part. Show your work.

  1. One of the 500 trips is picked at random. Given that it was taken by a casual rider, what is the probability that it was a long trip?
  2. For a trip picked at random from the 500, are the events "the trip is long" and "the trip was taken by a casual rider" independent? Back up your answer with numbers.
  3. Use your answer to part (a) as the chance that a casual rider's trip is long. Find the expected fee for one casual rider's trip, and say what that number means for the company.
  4. Suppose casual riders' trips are independent and each is long with probability 0.40. For 10 casual riders' trips, find the probability that at least 5 of them are long.

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Check yourself and see a model answer

Checklist

Our own checklist of what a full answer does, one line per point. Tick each one your response earns.

0 of 7 points

Model answer, annotated

(a) The condition is a casual rider, so the group is the 200 casual-rider trips. Of those, 80 were long, so P(long given casual) = 80/200 = 0.40.

  • Point 1 Uses the casual-rider total of 200 as the denominator, as the word given requires.

(b) Overall, P(long) = 120/500 = 0.24. But P(long given casual) = 0.40 from part (a). Knowing that a trip was taken by a casual rider raises the chance that it is long from 0.24 to 0.40, so the two events are not independent.

  • Point 2 Places the overall probability beside the conditional one, with both numbers shown.
  • Point 3 Draws the conclusion from the comparison and says what the difference means.

(c) Let F be the fee for one casual trip. F = 3 dollars with probability 0.60 and 7 dollars with probability 0.40, so E(F) = 0.60(3) + 0.40(7) = 1.80 + 2.80 = 4.60 dollars. Over many casual riders' trips, the company collects an average of about 4.60 dollars per trip; no single trip costs 4.60 dollars.

  • Point 4 Lists each fee with its probability, multiplies and adds.
  • Point 5 Calls the expected value a long-run average per trip, with the unit and the context.

(d) Let X be the number of long trips among the 10. Each trip is long or not, the trips are independent, n = 10 is fixed and p = 0.40 each time, so X is binomial with n = 10 and p = 0.40. Then P(X at least 5) = 1 - P(X at most 4) = 1 - 0.6331 = 0.3669, about 0.367.

  • Point 6 Names the model and its parameters and says why each binomial condition holds.
  • Point 7 Uses the complement to get the answer and rounds it sensibly.