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Statistics 9-12 / Week 10 / Thursday
4/6
Week 10 · Probability Rules

Thursday

Given that, and the test for independence
// The lane draw, with and, or and not
⏱ about 20 min

Thursday: Given That, and the Test for Independence

"Here is the sharper question," Comet says. "If I know the token is high, how likely is it even?"

"Then forget the low tokens," Wren says, covering them. "Only 3 outcomes are left: 6, 7, 8. Two are even."

"2/3," Comet says. "Higher than 1/2. Knowing high changed the chance."

"Now try low," Wren says. "2 even out of 4. That is 1/2, the same as P(even)."

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

"Divide probabilities," Wren says. "P(even and high) over P(high): 1/4 ÷ 3/8 = 2/3."

"What do you notice?" he adds. "Even and low are independent. Even and high are not."

Way 1: shrink the sample space

"Given that B happened" means only B's outcomes are possible now. B becomes the whole world.

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

For the crew: P(even given high) = 2 ÷ 3 = 2/3.

Way 2: divide the probabilities

P(A given B) = P(A and B) ÷ P(B). Both fractions have 8 underneath, so the 8s cancel.

For the crew: 1/4 ÷ 3/8 = 2/3 again. Two ways, one answer.

Turned around, the same equation gives a rule for "and": P(A and B) = P(B) × P(A given B).

Why the group total goes on the bottom

  1. Start with the whole sample space and count every outcome: that is the usual total.
  2. Hear "given B". Only B's outcomes could have happened, so cross out every outcome outside B.
  3. Count what is left: the outcomes in B. That count is the new total.
  4. Count the outcomes in A that survived: the outcomes in both A and B.
  5. Divide the second count by the first: P(A given B) = (both) ÷ (B).
WHY THE CONDITIONAL USES THE GROUP TOTAL, IN ORDER
  • ?Count the outcomes in both A and B
  • ?Divide: P(A given B) = (both) ÷ (B)
  • ?Count what is left: the outcomes in B become the new total
  • ?Count the whole sample space
  • ?Hear "given B" and cross out every outcome outside B
WHY THIS EXERCISE"Given" changes the total, not the overlap. That is the whole idea of conditional probability.

The test for independence

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.

PairP(A given B)P(A)P(A) × P(B)P(A and B)Independent?
even, given low1/21/21/2 × 1/2 = 1/41/4yes
even, given high2/31/21/2 × 3/8 = 3/161/4no

In everyday words: a low token is even just as often as any token. Low tells you nothing about even.

A high token is even more often than a token picked at random. High and even are linked in this bag.

GIVEN THAT
  • Read the question.
  • Tap your answer.
P(even and high) = 1/4 and P(high) = 3/8. What is P(even given high)?
P(low and even) = 1/4 and P(even) = 1/2. What is P(low given even)?
P(even) = 1/2, P(low) = 1/2 and P(even and low) = 1/4. Are the events independent?
P(even) = 1/2, P(high) = 3/8 and P(even and high) = 1/4. Are the events independent?
WHICH CONCLUSION IS JUSTIFIED?
  • Read the question.
  • Tap your answer.
P(even given high) = 2/3 and P(even) = 1/2. What follows?
P(high) = 3/8 and P(even given high) = 2/3. What is P(high and even)?
On the sign-up cards, P(stretched given one lap) = 3/5 and P(stretched) = 9/20. Independent?
The group named after "given" becomes the new what? Type one word.
Two events with P(A and B) = P(A) × P(B) are called what? Type one word.
StatementTrue or false?
1/2 × 1/2 = 1/4?
If P(A given B) = P(A), then A and B are independent.?
P(A given B) and P(B given A) are always equal.?
Linked events in a bag prove that one outcome causes the other.?
WHY THIS EXERCISEIndependence, linkage and cause are three different claims. The bag settles only the first two.
Draw the sample space twice. Cross out the low tokens in one copy and the high tokens in the other. Circle the even survivors.

Careful thinking. Tomorrow you put "given that" into everyday words with the sign-up cards and review the week.

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