TwiddleLab / Math / Probability by Simulation
Given That...
What you will learn
- A clue removes every outcome that doesn't fit it. The cells left are the new whole.
- Conditional probability: P(A | B) = cells in A and B / cells in B. Read it "A given B".
- A clue can make an event more likely (doubles given over 8: 2 of 10) or less likely.
How the lesson goes
Your friend rolled two dice. She says the total is over 8.
A double usually comes up 1 time in 6. What about now?
Use the clue: wipe every roll that doesn't fit.
Tap the clue. Then tap the doubles among the cells that are left.
Over 8 leaves totals 9, 10, 11, 12: 4 + 3 + 2 + 1 = 10 cells. Two of them are doubles (5 and 5, 6 and 6), so the chance is 2/10 = 0.2.
Build your own clue.
Make a clue where a double is at least 1 in 3, and at least 3 cells are left.
A clue that keeps few cells, with a double among them, makes the double likely. Total over 10, total is 10 or total under 5 all give exactly 1 in 3.
Given that...
A clue is an event you know happened. It wipes every outcome that doesn't fit.
Conditional probability is the chance inside what is left. Write P(A | B) and read it "A given B".
P(A | B) = cells in A and B / cells in B. Here: P(double | over 8) = 2 / 10.
Roll 2000 times with your clue.
Each roll lands in its cell. Rolls outside your clue drop out. Do doubles match your math among the rest?
Game: detective.
Two dice were rolled in secret. Ask yes or no questions to wipe cells, then tap the cell it is. Par is 5 moves.
Each answer is a clue that wipes the cells that don't fit. With k cells left, a guess is right with chance 1/k: a conditional probability that grows as the whole shrinks.
Two spam filters. Which one do you trust when it says spam?
100 emails, 20 are spam. Tap each model to see only the emails it flagged, then pick one.
Trust when it says spam means P(spam | flagged). A: 18 / 25 = 0.72. B: 12 / 13 = 0.92. The clue is "flagged", so you count only inside each model's pile.
Given a total of 7, paint "first die under 3".
Only the 6 cells with total 7 are left. Paint the ones that fit, then Check.
Total 7 leaves 6 cells. The first die is under 3 in two of them, 1 and 6 and 2 and 5, so P(first under 3 | total 7) = 2/6 = 1/3.
Speed round: four clues, 40 seconds.
Count only inside the clue. Tap the right chance. Get 3 of 4 for the badge.
Each answer is yellow cells over bright cells: 3/6, 2/6, 1/3 and 1/6.
Previous lesson: At Least One: The Complement Trick · Next lesson: The Long-Run Average · Part of the course: Probability by Simulation
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