You have two decks of cards: a 52 card deck (26 black, 26 red) and a 26 card deck (13 black, 13 red). You randomly draw two cards and win if both are the same color. Which deck would you prefer? What if the 26 card deck was randomly drawn from the 52 card deck? Which deck would you prefer then?
a deck of pokers. Three choices: A: 26 black, 26 Red; B: 13 black, 13 Red; C: random 26 card from the deck. Take the first two cards, if same color, the win $1, otherwise lose $1. Which deck is best for you if you are playing? Why? How to do simulations? How to draw the random pile of 26 cards?
Interesting question: From a deck of 52 cards pick 26 at random. From this set of 26 you pick two cards. You win if the both of these cards are of the same color. Is this a game you would prefer over one in which you win by picking two (first two picks) of the same color at random from a deck of 26 with equal number of black and red cards
toss 100 coins, what is the probability of getting more than 60 heads?
A tosses n+1 coins. B tosses n coins. B wins if he has at least as many heads as A. What is the probability that B wins?
You throw 1000 darts. Each one has a 50% chance to score. For the first 500 darts each is worth 1 point, for the second 500 darts each is worth 3 points. If you score 1500 points. Most likely how many 3point darts have you scored.
Basic linear algebra and basic calculus
Suppose you want to gamble in Vegas. In a game, you win $x if the number is prime and lose $x/2 if composite. The number is uniformly randomly generated by a machine between 1 and 10 inclusive. Will you play this game? Follow up: What if you can play n number of times and then stop. Will you play it?
You play rock, paper, scissors with an opponent, but your opponent cannot play rock. What should you play to maximize your expected profit if every time you win you win $1, lose you lose $1, draw you win $0?
Minimize the number of comparisons for finding minimum and maximum of a given set of numbers.
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