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Statistics 9-12 / Week 10 / Friday
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Week 10 · Probability Rules

Friday

Everyday given-thats
// The lane draw, with and, or and not
⏱ about 20 min

Friday: Everyday Given-Thats

"My cousin says most of the runners who stretched chose one lap," Comet says. "So stretching makes people pick one lap?"

"Careful," Wren says, spreading the sign-up cards. "That is P(one lap given stretched). You want P(stretched given one lap)."

"Different totals," Comet says. "One looks only at the stretchers. The other looks only at the one-lap runners."

Nova projects two shaded tables, a column lit in one, a row lit in the other. "Would you like a hint? Say each out loud."

"Of the stretchers, how many chose one lap. Of the one-lap runners, how many stretched," Comet says slowly.

"2/3 and 3/5," Wren says. "Everyday talk mixes them up all the time."

"Then our Race Day board needs clear labels," Comet says. "Every fraction says what it is given."

The sign-up cards

The crew sorted 40 sign-up cards by course and warm-up. Every count is the crew's own made-up tally.

stretchedjoggedTotal
one lap12820
two laps61420
Total182240
The sign-up cards: one lap or two laps against stretched or jogged, 40 runners.

Saying it in everyday words

"Given that" means "among only those". P(stretched given one lap) is "among the one-lap runners, the fraction who stretched".

Swapping the two events swaps the group. "Among the stretchers, the fraction who chose one lap" is a different question.

Independence in everyday words: "knowing the lane half tells you nothing about even". Linked: "one-lap runners stretched more often than runners in general".

Everyday sentenceProbability it means
Among the one-lap runners, three in five stretched.P(stretched given one lap) = 3/5
Among the stretchers, two in three chose one lap.P(one lap given stretched) = 2/3
Among the two-lap runners, three in ten stretched.P(stretched given two laps) = 3/10
Low or high, an even token came up just as often.P(even given low) = P(even): independent

Review: the week in five lines

  • P(A) = outcomes in A ÷ all outcomes.
  • P(not A) = 1 - P(A).
  • P(A or B) = P(A) + P(B) - P(A and B).
  • P(A given B) = outcomes in both ÷ outcomes in B = P(A and B) ÷ P(B).
  • Independent: P(A given B) = P(A), or P(A and B) = P(A) × P(B).
WHICH PROBABILITY IS IT?
  • Read the question.
  • Tap your answer.
"Among the two-lap runners, three in ten stretched." Which probability is this?
"Most of the stretchers chose one lap." Which probability is this?
"Knowing the lane half tells you nothing about even." Which word fits?
MIXED REVIEW
  • Read the question.
  • Tap your answer.
A two-way table: rows one lap and two laps, columns stretched and jogged, counts 12 and 8; 6 and 14, total 40.Given that a runner chose one lap, what is the probability the runner stretched?
A two-way table: rows one lap and two laps, columns stretched and jogged, counts 12 and 8; 6 and 14, total 40.Given that a runner stretched, what is the probability the runner chose one lap?
A two-way table: rows one lap and two laps, columns stretched and jogged, counts 12 and 8; 6 and 14, total 40.What is the probability a random runner chose two laps or jogged (or both)?
A two-way table: rows one lap and two laps, columns stretched and jogged, counts 12 and 8; 6 and 14, total 40.What is the probability a random runner did not stretch?
P(stretched given one lap) is 12 over 20. Write it in lowest terms as a fraction.
WHY THIS EXERCISEA probability is a fraction of a group. Lowest terms makes two fractions easy to compare.
StatementTrue or false?
"Among the stretchers" means the stretched column is the total.?
P(A given B) and P(B given A) always share the same top number.?
12 ÷ 18 = 2/3?
Most stretchers choosing one lap proves that stretching makes people choose one lap.?
WHY THIS EXERCISEEveryday claims hide a "given". Finding it is the difference between reading a table and guessing.
Try it
Say each row of the everyday-sentence table out loud, then say which group is the total.
Write one everyday sentence of your own and translate it into a probability.
Draw the sign-up table twice. Shade the one-lap row in one copy and the stretched column in the other. Circle the shared box.

A clear-headed week. Tomorrow is Track Day: a family draw from a bag of six.

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