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Gambling at Monte Carlo

A man went to Monte Carlo to try and make his fortune. Whilst he was there he had an opportunity to bet on the outcome of rolling dice. He was offered the same odds for each of the following outcomes: At least 1 six with 6 dice. At least 2 sixes with 12 dice. At least 3 sixes with 18 dice.

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Balls and Bags

Two bags contain different numbers of red and blue balls. A ball is removed from one of the bags. The ball is blue. What is the probability that it was removed from bag A?

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Coin Tossing Games

You and I play a game involving successive throws of a fair coin. Suppose I pick HH and you pick TH. The coin is thrown repeatedly until we see either two heads in a row (I win) or a tail followed by a head (you win). What is the probability that you win?

Card Game (a Simple Version of Clock Patience)

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

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Rules

Four cards (two cards with 1 and two with 2) are shuffled and placed into two piles of two

You have two trays labelled 1 and 2.

Starting with the first pile of cards - turn a card over:

  • If the card has a 1 on it - place it in tray 1 and take a card from pile 1
  • If the card has a 2 on it - place it in tray 2 and turn over the next card in pile 2.

You win if all your cards end up in the trays before you run out of cards in one pile.

You lose if there are cards left in a pile that you cannot reach.

What is the probability that you win?

What happens if you have six cards in the two piles (three with one and three with 2)?

Do your odds increase or decrease?

Can you see any patterns in successful and unsuccessful placement of cards in the piles?