# Probabilities in a card game.

Carol and David decide to play a game as follows:

Carol draws and keeps a card from a shuffled pack number 1 to 6.

David the draws a card from the remaining 5.

The winner is the one holding the card with the highest number.

a) Determine whether or not there is an advantage to drawing the first.

If the rules are now altered so that Carol returns her card to the pack after noting its value, and also, if Carol and David draw the same card all the cards are shuffled and they draw again.

Find the probability that:

b) Carol wins on the first draw

c) Carol wins on the forth draw

d) Carol wins, if the games can continue for as long as is necessary.

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#### Solution Preview

Carol draws a card from 1 to 6 with a probability of 1/6 Lets calculate what are David's chances to win:

If carol draws "1" than David wins for sure any card he draws will win).

The probability of this scenario is therefore

1/6 x 1 = 1/6

Carol draws "2" than David wins if he draws only one of 4 cards out of the 5

(3,4,5,6):

The probability of this scenario is therefore

1/6 x 4/5=4/30

Carol draws "3" than David wins if he draws only one of 3 cards out of the 5

(4,5,6):

The probability of this scenario is therefore

1/6 x 3/5=3/30

Carol draws "4" than David wins if he draws only ...