Question

We draw one card at a time without replacement from the top of a shuffled standard poker deck and stop when we draw an ace. Let X be the number of cards we have drawn, what is the P(X=5) and (P<5)

Answer 1

According to the given information , we draw one card at a time without replacement from the top of a shuffled standard poker deck and stop when we draw an ace. As we know in a shuffled standard poker deck, 4 ace card and rest 48 non ace card out off 52 cards.

Let X be the number of cards we have drawn then the probability that

1)

which is determined as 1st drawn non ace from 48 non ace card out of 52 cards  , 2nd drawn non ace from 47 non ace card out of 51 cards, 3rd drawn non ace from 46 non ace card out of 50 cards , 4th drawn non ace from 45 non ace card out of 49 cards and 5th drawn ace from 4  ace card out of 48 cards.

  

Therefore required probability is 0.0599

ii)

That implies the ace is drawn with in first 4 drawn ,which is equal to 1-probability that non ace is drawn with in first four drawn.

Therefore for first 4 drawn of non ace is calculated with without replacement as:

1st drawn non ace from 48 non ace card out of 52 cards  , 2nd drawn non ace from 47 non ace card out of 51 cards, 3rd drawn non ace from 46 non ace card out of 50 cards , 4th drawn non ace from 45 non ace card out of 49 cards

  

  

Therefore required probability is