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Lucky Guess

Problem

Jane shows John four playing cards: Ace of Spades, Two of Clubs, Three of Diamonds, and Four of Hearts. She shuffles the cards and places them face down on a table.

"I would like you to select two cards at random. If you select two cards of the same colour, I'll give you £1, if they're different, you give me £1. As there are two possible outcomes, we both stand an equal chance of winning £1." suggests Jane with a cheeky smile.

John's friend, James, secretly advises John, "Actually, there are three possible outcomes: black and red, black and black, red and red. As two of these outcomes is a win for you, I"d go for it!"

By finding the actual probability of John winning, show that neither Jane nor James are correct.


Solution

The probability of picking two reds, P(RR) = (2/4)(1/3) = 2/12; similarly P(BB) = 2/12.

Therefore, P(same colour) = 4/12 = 1/3.

Alternatively, it doesn't matter whether the first card is red or black,
P(2nd card is the same) = 1/3.

What if there was one red and three black cards?

Problem ID: 138 (Dec 2003)     Difficulty: 2 Star

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