## 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?