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Algebra 2 9-12 / Week 11 / Tuesday
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Week 11 · Probability in the Seed Trays

Tuesday

Given that: conditional probability
// Sprouted or not, misted or not
⏱ about 20 min

Tuesday: Given That, Conditional Probability

"Here is the real question," Comet says. "If I know a cell was misted daily, how likely is a sprout?"

"Then forget the weekly row," Wren says, covering it with his hand. "Only 24 cells are left. 18 sprouted."

"3/4," Comet says. "Higher than the whole tray. What do you notice if we cover the daily row instead?"

Wren slides his hand. "Weekly: 8 of 24. 1/3. Daily misting helped, in our tray."

Nova projects both fractions side by side. "Would you like a hint? There is a second way to get the same number."

"Divide probabilities instead of counts," Wren says. "P(sprouted and daily) over P(daily). 3/8 over 1/2."

"Same 3/4," Comet says. "Two ways, one answer. Given that is a powerful phrase."

Way 1: shrink the sample space

"Given that B happened" means only B's cells are possible now. The row or column for B becomes the whole world.

P(A given B) = the number of cells in both A and B, divided by the number of cells in B.

For the crew: P(sprouted given daily) = 18 ÷ 24 = 3/4.

Way 2: divide the probabilities

P(A given B) = P(A and B) ÷ P(B). Both fractions have the same total underneath, so the totals cancel.

For the crew: P(sprouted and daily) = 3/8 and P(daily) = 1/2. Dividing gives 3/4 again.

The same equation turned around gives a rule for "and": P(A and B) = P(B) × P(A given B).

Order matters

P(sprouted given daily) asks about the daily row: 3/4.

P(daily given sprouted) asks about the sprouted column: 18 of 26 = 9/13.

Same box on top, different group underneath. Read the "given" carefully every time.

QuestionGroup used as the totalCount in bothProbability
P(sprouted given daily)daily row, 24183/4
P(sprouted given weekly)weekly row, 2481/3
P(daily given sprouted)sprouted column, 26189/13
P(sprouted)whole tray, 482613/24
GIVEN THAT
  • Read the question.
  • Tap your answer.
A two-way table: rows Misted daily and Misted weekly, columns Sprouted and Did not sprout, counts 18 and 6; 8 and 16, total 48.Given that a cell is "Misted daily", what is the probability it is "Sprouted"?
A two-way table: rows Misted daily and Misted weekly, columns Sprouted and Did not sprout, counts 18 and 6; 8 and 16, total 48.Given that a cell is "Misted weekly", what is the probability it is "Sprouted"?
A two-way table: rows Misted daily and Misted weekly, columns Sprouted and Did not sprout, counts 18 and 6; 8 and 16, total 48.Given that a cell is "Sprouted", what is the probability it is "Misted daily"?
A two-way table: rows Misted daily and Misted weekly, columns Sprouted and Did not sprout, counts 18 and 6; 8 and 16, total 48.Given that a cell is "Misted weekly", what is the probability it is "Did not sprout"?
TWO WAYS, ONE ANSWER
  • Read the question.
  • Tap your answer.
P(sprouted and daily) = 3/8 and P(daily) = 1/2. What is P(sprouted given daily)?
P(daily) = 1/2 and P(sprouted given daily) = 3/4. What is P(sprouted and daily)?
Of the 24 cells misted weekly, how many sprouted? Type the number.
WHY THIS EXERCISEThat one count over the row total is P(sprouted given weekly). Conditional probability starts with the right box.
StatementTrue or false?
P(A given B) uses B's count as the total, not the whole tray.?
18 ÷ 24 = 3/4?
P(sprouted given daily) and P(daily given sprouted) are always equal.?
P(A and B) = P(B) × P(A given B).?
WHY THIS EXERCISEReading the "given" correctly is the whole skill. The arithmetic is just a fraction.
Try it
Cover the weekly row of your table with a card. Read the daily probabilities with only the uncovered cells.
Then cover the "Did not sprout" column and read P(daily given sprouted) the same way.
Draw the table twice. Shade the daily row in one copy and the sprouted column in the other. Circle the shared box.

Two ways that agree. Tomorrow is Lab day: a carton, a coin and a handful of beans build a tray of your own.

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