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Algebra 2 9-12 / Week 11 / Thursday
4/6
Week 11 · Probability in the Seed Trays

Thursday

The test for independence
// Sprouted or not, misted or not
⏱ about 20 min

Thursday: The Test for Independence

"Second tray," Wren says, chalking a new grid. "Sunny bench against shaded bench. Sprouted or not."

Comet reads the counts. "12, 8, 6, 4. What do you notice?"

"P(sprouted) is 3/5," Wren says. "P(sprouted given sunny) is 12 of 20. Also 3/5."

"And given shaded, 6 of 10. 3/5 again," Comet says. "Knowing the bench tells us nothing."

Nova projects the two trays side by side. "Would you like a hint? Try multiplying P(A) by P(B) in each tray."

"Sunny tray: 3/5 × 2/3 = 2/5. And P(sprouted and sunny) is 2/5. Equal," Wren says.

"Misting tray: 13/24 × 1/2 = 13/48, but P(sprouted and daily) is 3/8. Not equal. Misting mattered, the bench did not."

Two tests, one idea

Events A and B are independent when knowing B happened does not change the chance of A.

Test 1: compare P(A given B) with P(A). Equal means independent.

Test 2: compare P(A and B) with P(A) × P(B). Equal means independent.

The two tests always agree, because P(A given B) = P(A and B) ÷ P(B). If P(A given B) = P(A), then P(A and B) = P(A) × P(B).

The crew's two trays

SproutedDid not sproutTotal
Sunny bench12820
Shaded bench6410
Total181230
The bench tray: sunny or shaded against sprouted or not, 30 cells, built so the bench makes no difference.
TrayP(A) × P(B)P(A and B)Independent?
Bench: sprouted and sunny3/5 × 2/3 = 2/52/5yes
Misting: sprouted and daily13/24 × 1/2 = 13/483/8no

In everyday words: on the bench tray, a sunny cell sprouts just as often as any cell. The bench is not linked to sprouting.

On the misting tray, a daily-misted cell sprouts more often than a cell picked at random. Misting and sprouting are linked.

A link in a table is not a cause by itself. The crew assigned the misting, so here it is fair to say misting helped.

TEST FOR INDEPENDENCE
  • Read the question.
  • Tap your answer.
P(Sprouted) = 3/5 and P(Sprouted given Sunny bench) = 3/5. Are "Sprouted" and "Sunny bench" independent?
P(Sprouted) = 13/24 and P(Sprouted given Misted daily) = 3/4. Are "Sprouted" and "Misted daily" independent?
On the bench tray, P(sprouted) = 3/5 and P(sunny) = 2/3. What is P(sprouted) × P(sunny)?
On the misting tray, P(sprouted) × P(daily) = 13/48 but P(sprouted and daily) = 3/8. What does this show?

Why the addition rule subtracts

  1. Count the cells in A.
  2. Count the cells in B.
  3. Adding the two counts counts every cell in both A and B twice.
  4. Subtract the overlap once, so every cell in A or B is counted exactly once.
  5. Divide by the total: P(A or B) = P(A) + P(B) - P(A and B).
THE ADDITION RULE, IN ORDER
  • ?Divide by the total to get P(A or B) = P(A) + P(B) - P(A and B)
  • ?Count the cells in B
  • ?Subtract the overlap once so each cell counts once
  • ?Adding the two counts counts the overlap twice
  • ?Count the cells in A
WHY THIS EXERCISEThe minus sign in the addition rule is not a trick. It repairs a double count.
The cells counted twice when you add two events are the what? Type one word.
Two events with P(A and B) = P(A) × P(B) are called what? Type one word.
StatementTrue or false?
If P(A given B) = P(A), then A and B are independent.?
3/5 × 2/3 = 2/5?
A link between two events in a table proves one caused the other.?
P(A or B) = P(A) + P(B) for every pair of events.?
WHY THIS EXERCISEIndependence, linkage and cause are three different claims. The table settles only the first two.
Try it
Return to your lab carton table. Multiply P(flag) by P(bean) and compare with P(flag and bean).
Write how far apart they are. A small carton usually misses by a little.
Draw two overlapping circles labeled A and B. Shade the overlap and write the addition rule under them.

Careful thinking. Tomorrow you put conditional probability into everyday words and review the whole week.

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