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Statistics 9-12 / Week 10 / Tuesday
2/6
Week 10 · Probability Rules

Tuesday

The addition rule, two ways
// The lane draw, with and, or and not
⏱ about 20 min

Tuesday: The Addition Rule, Two Ways

"Even or low came out 3/4 yesterday," Comet says. "I listed every token. Is there a faster way?"

"Add the two probabilities," Wren says. "1/2 plus 1/2 is 1. But that cannot be right."

"Because tokens 2 and 4 got counted twice," Comet says. "They are in both loops."

"So subtract them once," Wren says. "1/2 + 1/2 - 1/4 = 3/4. Same answer as your list."

Nova projects two overlapping loops with the shared part glowing. "Would you like a hint? The glow is the overlap."

"P(A or B) = P(A) + P(B) - P(A and B)," Wren writes. "The addition rule."

"Two ways, one answer," Comet says. "Listing when the set is small, the rule when it is not."

"What do you notice when two events share no outcome?" Wren asks. "Then there is nothing to subtract."

Way 1: list the union

Write every outcome in A, then add the outcomes of B that are not already there. Count, then divide by 8.

Even or high: {2, 4, 6, 8} plus the new high outcome 7 gives {2, 4, 6, 7, 8}. So P = 5/8.

Way 2: the addition rule

P(A or B) = P(A) + P(B) - P(A and B). Adding P(A) and P(B) counts the overlap twice, so subtract it once.

Even or high: 1/2 + 3/8 - 1/4 = 5/8. The rule agrees with the list.

When A and B share no outcome, P(A and B) = 0 and the rule is just P(A) + P(B). Such events are called disjoint.

QuestionP(A)P(B)P(A and B)P(A or B)
even or low1/21/21/43/4
even or high1/23/81/45/8
odd or even1/21/201
low or high1/23/807/8

"Odd or even" and "low or high" are disjoint pairs: nothing to subtract. Lane 5 is in neither low nor high, so the last row is not 1.

USE THE ADDITION RULE
  • Read the question.
  • Tap your answer.
P(even) = 1/2, P(low) = 1/2 and P(even and low) = 1/4. What is P(even or low)?
P(even) = 1/2, P(high) = 3/8 and P(even and high) = 1/4. What is P(even or high)?
P(low) = 1/2, P(high) = 3/8 and P(low and high) = 0. What is P(low or high)?
Tokens 1 to 8 are equally likely. Odd = {1, 3, 5, 7} and low = {1, 2, 3, 4}. What is P(odd or low)?
TWO WAYS, ONE ANSWER
  • Read the question.
  • Tap your answer.
By listing, even or high is {2, 4, 6, 7, 8}. What is its probability?
Two events share no outcome. What does the addition rule become?
Why is 1/2 + 1/2 not the probability of even or low?
Events that share no outcome are called what? Type one word.
WHY THIS EXERCISEDisjoint events are the one case where adding probabilities needs no correction.
StatementTrue or false?
1/2 + 3/8 - 1/4 = 5/8?
P(A or B) can be larger than 1 if A and B are big enough.?
For disjoint events, P(A or B) = P(A) + P(B).?
1/2 + 3/8 = 7/8?
WHY THIS EXERCISEThe minus sign in the addition rule repairs a double count. It is not a trick.
Try it
Pick two events of your own from the lane bag, such as "3 or less" and "a multiple of 3".
Find P(A or B) by listing, then by the rule. The two numbers must match.
Draw two overlapping loops labeled even and high. Write the tokens in each part and the rule underneath.

Two ways that agree. Tomorrow is Data Lab: forty real draws from the bag, tallied two ways at once.

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