A game is played with a deck of ten cards numbered from to . Shuffle the deck thoroughly. Take the top card. If it is numbered , you win. If it is numbered , where , then replace the card into the th position from the top and draw again. You are allowed a maximum of draws before losing the game. What is the probability of winning?

Source: NCTM Mathematics Teacher, September 2006

**Solution**

It helps to build a deck of cards made out of cutout papers and simulate a few draws in order to understand the mechanics of the game. For example, we discover that having the card numbered in third position does not guarantee a win, unless the card numbered is not in first position.

An indirect approach to solve the problem is to figure out how many ways can we lose the game? If the card numbered is in positions , or from the top, we will definitely lose because there is no way for it to bubble up to the top in draws. The probability for each of these seven events equals . We will also lose if the card numbered is in third position and the card numbered is in first position. The probability for this event equals .

The probability for losing equals

Hence, the probability for winning equals

**Answer**:

**Alternative solution**

A direct approach to solve the problem is to figure out how many ways can we win the game? If the card numbered is in first or second positions, we will win for sure. The probability for each of these two events equals . If the card numbered is in third position and the card numbered is not in first position, we will also win. The probability for this event equals .

The probability of winning equals