## Probability

An archive of questions and answers that may be of interest to puzzle enthusiasts.
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Question 1 - amoeba:
A jar begins with one amoeba. Every minute, every amoeba turns into 0, 1, 2, or 3 amoebae with probability 25% for each case ( dies, does nothing, splits into 2, or splits into 3). What is the probability that the amoeba population eventually dies out? Show Answer

Question 2 -apriori:
An urn contains one hundred white and black balls. You sample one hundred balls with replacement and they are all white. What is the probability that all the balls are white? Show Answer

Question 3 - bayes:
One urn contains black marbles, and the other contains white or black marbles with even odds. You pick a marble from an urn; it is black; you put it back; what are the odds that you will draw a black marble on the next draw? What are the odds after n black draws? Show Answer

Question 4 - birthday/line:
At a movie theater, the manager announces that they will give a free ticket to the first person in line whose birthday is the same as someone who has already bought a ticket. You have the option of getting in line at any time. Assuming that you don't know anyone else's birthday, that birthdays are distributed randomly throughtout the year, etc., what position in line gives you the greatest chance of being the first duplicate birthday? Show Answer

Question 5 - birthday/same.day:
How many people must be at a party before you have even odds or better of two having the same bithday (not necessarily the same year, of course)? Show Answer

Question 6 - cab:
A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. Here is some data:

a) Although the two companies are equal in size, 85% of cab accidents in the city involve Green cabs and 15% involve Blue cabs.

b) A witness identified the cab in this particular accident as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.

What is the probability that the cab involved in the accident was Blue rather than Green?

If it looks like an obvious problem in statistics, then consider the following argument:

The probability that the color of the cab was Blue is 80%! After all, the witness is correct 80% of the time, and this time he said it was Blue!

What else need be considered? Nothing, right?

If we look at Bayes theorem (pretty basic statistical theorem) we should get a much lower probability. But why should we consider statistical theorems when the problem appears so clear cut? Should we just accept the 80% figure as correct? Show Answer

Question 7 - coupon:
There is a free gift in my breakfast cereal. The manufacturers say that the gift comes in four different colors, and encourage one to collect all four (& so eat lots of their cereal). Assuming there is an equal chance of getting any one of the colors, what is the expected number of boxes I must consume to get all four? Can you generalise to n colors and/or unequal probabilities? Show Answer

Question 8 - darts:
Peter throws two darts at a dartboard, aiming for the center. The second dart lands farther from the center than the first. If Peter now throws another dart at the board, aiming for the center, what is the probability that this third throw is also worse (i.e., farther from the center) than his first? Assume Peter's skilfulness is constant. Show Answer

Question 9 - derangement:
12 men leave their hats with the hat check. If the hats are randomly returned, what is the probability that nobody gets the correct hat? Show Answer

Question 10 - family:
Suppose that it is equally likely for a pregnancy to deliver a baby boy as it is to deliver a baby girl. Suppose that for a large society of people, every family continues to have children until they have a boy, then they stop having children. After 1,000 generations of families, what is the ratio of males to females? Show Answer

Question 11 - flips/once.in.run:
What are the odds that a run of one H or T (i.e., THT or HTH) will occur in n flips of a fair coin? Show Answer

Question 12 - twice.in.run:
What is the probability in n flips of a fair coin that there will be two heads in a row? Show Answer

Question 13 - flips/unfair:
Generate even odds from an unfair coin. For example, if you thought a coin was biased toward heads, how could you get the equivalent of a fair coin with several tosses of the unfair coin? Show Answer

Question 14 - flips/waiting.time:
Compute the expected waiting time for a sequence of coin flips, or the probabilty that one sequence of coin flips will occur before another. Show Answer

Quesition 15 - flush:
Which set contains proportionately more flushes than the set of all possible poker hands? (1) Hands whose first card is an ace (2) Hands whose first card is the ace of spades (3) Hands with at least one ace (4) Hands with the ace of spades Show Answer

Question 16 - hospital:
A town has two hospitals, one big and one small. Every day the big hospital delivers 1000 babies and the small hospital delivers 100 babies. There's a 50/50 chance of male or female on each birth. Which hospital has a better chance of having the same number of boys as girls? Show Answer

Question 17 - icos:
The "house" rolls two 20-sided dice and the "player" rolls one 20-sided die. If the player rolls a number on his die between the two numbers the house rolled, then the player wins. Otherwise, the house wins (including ties). What are the probabilities of the player winning? Show Answer

Question 18 - intervals:
Given two random points x and y on the interval 0..1, what is the average size of the smallest of the three resulting intervals? Show Answer

Question 19 - killers.and.pacifists:
You enter a town that has K killers and P pacifists. When a pacifist meets a pacifist, nothing happens. When a pacifist meets a killer, the pacifist is killed. When two killers meet, both die. Assume meetings always occur between exactly two persons and the pairs involved are completely random. What are your odds of survival? Show Answer

Question 20 - leading.digit:
What is the probability that the ratio of two random reals starts with a 1? What about 9? Show Answer

Question 21 - lights:
Waldo and Basil are exactly m blocks west and n blocks north from Central Park, and always go with the green light until they run out of options. Assuming that the probability of the light being green is 1/2 in each direction, that if the light is green in one direction it is red in the other, and that the lights are not synchronized, find the expected number of red lights that Waldo and Basil will encounter. Show Answer

Question 22 - lottery:
There n tickets in the lottery, k winners and m allowing you to pick another ticket. The problem is to determine the probability of winning the lottery when you start by picking 1 (one) ticket.

A lottery has N balls in all, and you as a player can choose m numbers on each card, and the lottery authorities then choose n balls, define L(N,n,m,k) as the minimum number of cards you must purchase to ensure that at least one of your cards will have at least k numbers in common with the balls chosen in the lottery. Show Answer

Question 23 - oldest.girl:
You meet a stranger on the street, and ask how many children he has. He truthfully says two. You ask "Is the older one a girl?" He truthfully says yes. What is the probability that both children are girls? What would the probability be if your second question had been "Is at least one of them a girl?", with the other conditions unchanged? Show Answer

Question 24 - particle.in.box:
A particle is bouncing randomly in a two-dimensional box. How far does it travel between bounces, on average?

Suppose the particle is initially at some random position in the box and is traveling in a straight line in a random direction and rebounds normally at the edges. Show Answer

Question 25 - pi:
Are the digits of pi random (i.e., can you make money betting on them)? Show Answer

Question 26 - random.walk:
Waldo has lost his car keys! He's not using a very efficient search; in fact, he's doing a random walk. He starts at 0, and moves 1 unit to the left or right, with equal probability. On the next step, he moves 2 units to the left or right, again with equal probability. For subsequent turns he follows the pattern 1, 2, 1, etc.

His keys, in truth, were right under his nose at point 0. Assuming that he'll spot them the next time he sees them, what is the probability that poor Waldo will eventually return to 0? Show Answer

Question 27 - reactor:
There is a reactor in which a reaction is to take place. This reaction stops if an electron is present in the reactor. The reaction is started with 18 positrons; the idea being that one of these positrons would combine with any incoming electron (thus destroying both). Every second, exactly one particle enters the reactor. The probablity that this particle is an electron is 0.49 and that it is a positron is 0.51.

What is the probability that the reaction would go on for ever?

Note: Once the reaction stops, it cannot restart. Show Answer

Question 28 - roulette:
You are in a game of Russian roulette, but this time the gun (a 6 shooter revolver) has three bullets _in_a_row_ in three of the chambers. The barrel is spun only once. Each player then points the gun at his (her) head and pulls the trigger. If he (she) is still alive, the gun is passed to the other player who then points it at his (her) own head and pulls the trigger. The game stops when one player dies.

Now to the point: would you rather be first or second to shoot? Show Answer

Question 29 - transitivity:
Can you number dice so that die A beats die B beats die C beats die A? What is the largest probability p with which each event can occur? Show Answer

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