Probability paradoxes: Monty Hall, the birthday problem, and more

Questions

1. 1
   Not answered. What range of numbers can a probability take?
2. 2
   Not answered. A weather model gives a 30% chance of rain tomorrow. What is the chance of no rain?
3. 3
   Not answered. When every outcome in the sample space is equally likely, how do you find the probability of an event?
4. 4
   Not answered. You roll two fair six-sided dice and add them. Which total is the most likely?
5. 5
   Not answered. Two events are independent. How do you find the probability that both happen?
6. 6
   Not answered. On one roll of a fair die, what is the probability of rolling a 1 or a 2?
7. 7
   Not answered. What does the conditional probability P(A | B) measure?
8. 8
   Not answered. What is the expected (average) value of one roll of a fair six-sided die?
9. 9
   Not answered. You roll one die four times. Is getting at least one six more likely than not?
10. 10
    Not answered. A fair coin has landed heads five times in a row. What is the probability the next flip is heads?
11. 11
    Not answered. Which belief is the gambler's fallacy?
12. 12
    Not answered. Why doesn't a streak of heads make tails more likely on the next flip?
13. 13
    Not answered. At the Monte Carlo Casino in 1913, the roulette ball reportedly landed on black 26 times in a row. What happened next?
14. 14
    Not answered. What does the law of large numbers actually promise about flipping a fair coin?
15. 15
    Not answered. As you keep flipping a fair coin, what happens to the raw gap between the number of heads and the number of tails?
16. 16
    Not answered. Someone walks in, sees you roll a double six, and says, "You must have been rolling for a long time to finally get that." What mistake is this?
17. 17
    Not answered. The gambler's fallacy expects a streak to reverse. Which belief expects a streak to continue?
18. 18
    Not answered. Monty Hall: three doors hide a car and two goats. You pick a door. The host, who knows where the car is, opens a different door to show a goat and offers you the chance to switch. Should you?
19. 19
    Not answered. In the standard Monty Hall game, what makes switching better than staying?
20. 20
    Not answered. The 2/3 advantage for switching depends on how the host behaves. Which rules does it need?
21. 21
    Not answered. Now imagine 100 doors. You pick one, and the host opens 98 of the others, all goats, leaving one door closed. Should you switch?
22. 22
    Not answered. Variant: the host forgets where the car is and opens one of the other two doors at random. It happens to show a goat. Should you switch now?
23. 23
    Not answered. When Marilyn vos Savant published the "switch" answer in Parade magazine in 1990, how did readers react?
24. 24
    Not answered. Where did the Monty Hall problem first appear in print in its game-show form?
25. 25
    Not answered. How many people need to be in a room before it's more likely than not that two of them share a birthday?
26. 26
    Not answered. Why are just 23 people enough to make a shared birthday likely?
27. 27
    Not answered. What's the easiest way to calculate the probability of at least one shared birthday?
28. 28
    Not answered. In a room of 70 people, roughly how likely is it that two share a birthday?
29. 29
    Not answered. Ignoring leap years, so there are 365 possible birthdays, how many people guarantee that two of them share a birthday?
30. 30
    Not answered. How many other people need to be in a room before there's a better-than-even chance that someone shares your birthday?
31. 31
    Not answered. The "birthday attack" in cryptography uses the same math. Roughly how many random items do you need to generate before you're likely to find a collision among N equally likely hash values?
32. 32
    Not answered. A disease affects 1% of people. A test is 99% accurate for both sick and healthy people. You test positive. What's the chance you actually have the disease?
33. 33
    Not answered. Linda is 31, single, outspoken, and studied philosophy. She cares deeply about social justice. Which is more probable?
34. 34
    Not answered. What is Simpson's paradox?
35. 35
    Not answered. A family has two children. You learn that at least one is a boy, but not which one. If boys and girls are equally likely and independent, what's the probability that both are boys?
36. 36
    Not answered. In many real-world datasets, like populations, street addresses, and financial figures, how often is the first digit a 1?
37. 37
    Not answered. A game: flip a fair coin until it lands heads. If the first head comes on flip k, you win 2^k dollars. What's the expected payout?
38. 38
    Not answered. On an American roulette wheel with 38 pockets, you bet $1 on a single number, which pays 35 to 1. What do you expect to win or lose per $1 bet in the long run?