# Card Games: Play Your Cards Right

This problem is based on the card game Play Your Cards Right, otherwise known as the Higher or Lower game.

Rules: A standard French deck of 52 playing cards is shuffled and 9 cards are placed face down. The first card is turned over. The player must guess whether the next card’s value will be higher or lower than the previous card’s value. Assume that the ace is the lowest card in the pack (i.e. has a value of 1).

If the next card turns out to have the exact same value (for example, the player turns over a king followed by another king), the player loses-since this is neither higher nor lower.

The player wins if he has turned over all of the cards, having correctly guessed whether each of the eight is higher or lower than the preceding one.

If you want to have a go at playing it go ahead and try it here. In this problem we shall be using a full deck of cards.

Let's demonstrate with an example. You pick 9 cards from a full standard deck (no Jokers). The first card is a

.

We know that there are more cards that are higher than the 6 of clubs, so you say that the next card will be higher. The next card is turned over.

It's lower! You lose.

The probability of winning can be expressed as $$x\%$$. Find $$x$$. (It is a decimal).

Details and Assumptions:

• Our deck contains, in ascending order of value, aces, twos, threes, fours, fives, sixes, sevens, eights, nines, tens, jacks, queens, kings. Each of these occurs four times.

• Assume that you play optimally.