1020 ppt
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This document discusses the famous Monty Hall problem, which is a probability puzzle involving a game show with 3 doors - behind one door is a car, behind the others are goats. The host, who knows what's behind each door, opens one door to reveal a goat after the player picks a door. Players are then asked if they want to switch to the other unopened door. The document analyzes this problem through different cases and arguments, and concludes that the optimal strategy is for the player to switch doors, as the probability of picking the correct door initially is 1/3, so the other unopened door has a 2/3 probability of containing the car.
1020 ppt
- 1. Haifeng Luo
- 2. The Monty Hall Problem Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?”
- 4. Background First appeared in 1975, name comes from a TV show. Original author claimed that you should switch. Thousands of people disagreed, including many PhDs.
- 5. Two arguments It is optimal to switch: probability of picking the correct door initially is 1/3. So the other door has 2/3. Or It does not matter: the remaining two doors are equally likely to contain the car.
- 6. Answer: SWITCH! The key lies in how the host makes his decision. If he intentionally opens a door with goat, then no information is gain. If the door is randomly chosen, then the two doors have equal probability of containing the car.
- 7. Case 1 Player picks 1 1/3 1/3 1/3 Car @ 1 Car @ 2 Car @ 3 Host shows 2 or 3 Host shows 3 Host shows 2 Host always shows goat!
- 8. Case 2 Player picks 1 1/3 1/3 1/3 1/2 1/2 Shows 2 -- goat Shows 3 -- goat Shows 2 -- car Shows 3 -- goat Shows 2 -- goat Shows 3 -- car Car @ 1 Car @ 2 Car @ 3 ✔ ✔ ✖ ✖ Each case has 1/6 Probability
- 9. Back to Case 1 Player picks 1 1/3 1/3 1/3 1/2 1/2 Shows 2 -- goat Shows 3 -- goat Shows 2 -- car Shows 3 -- goat Shows 2 -- goat Shows 3 -- car Car @ 1 Car @ 2 Car @ 3 ✔ ✔ ✖ ✖ 1/6 1/6 0 1/3 1/3 0
- 10. Boys and Girls Suppose a society really prefers boys over girls. Each family tries their best to have a boy to continue the male line. Each couple will have one baby per year, and they stop once they get a boy. A typical family may have, say, g, g, g, g, b. What’s the percentage of boys/girls after 100 years?
- 11. Still 50%! Let’s say we have 1000 couples in the society. 1st year: 500 boys; 500 girls 2nd year: from the frustrated couples -- 250 boys; 250 girls 3rd year: as they keep going: 125 boys and 125 girls And it continues …
- 12. The take-away Our intuitions can be very misleading. Especially regarding probability and statistics. For more interesting examples, refer to the book Thinking, fast and slow by Daniel Kahneman.