Thread: A game is played with a deck of ten cards...

A game is played with a deck of ten cards numbered from 1 to 10. Shuffle the deck thoroughly. Take the top card. If it is numbered one, you win. If it in numbered k, where k>1, then replace the card into the kth positition from the top and draw again. You are allowed a maximum of 3 draws before losing the game. What is the probability of winning?

After the 1/10 chance on the first draw, I no longer know how to solve.

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Originally Posted by ceasar_19134

A game is played with a deck of ten cards numbered from 1 to 10. Shuffle the deck thoroughly. Take the top card. If it is numbered one, you win. If it in numbered k, where k>1, then replace the card into the kth positition from the top and draw again. You are allowed a maximum of 3 draws before losing the game. What is the probability of winning?

After the 1/10 chance on the first draw, I no longer know how to solve.

So, if a 1 is on top you win, chance of this happening: 1/10

If a 2 is on top, then you will only win if the second card is a 1, chance of this happening: 1/10*1/9=1/90

If anything else is on top, you will only win if 1 is either the second or third card, chance of this happening is:
(chance of second card being 1) + (chance of third card being one)
= (8/10 * 1/9) + (8/10 * 7/9 * 1/8)
= 8/90 + 7/90
= 15/90

Add all those together to get:
9/90+1/90+15/90=25/90=5/14

But make sure to CHECK MY ARITHMETIC!!!!!!!!!!!