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I draw cards from a deck until i get a spade.

1) What is the probability that i need exactly 7 draws?

2) Given that six or more draws are required, what is the probability that exactly seven draws are required?

May 1, 2015

#3
+27902
+5

1.  There are 52 cards in a pack.  13 of them are spades and the other 39 are not.

For there to be exactly seven draws before you get a spade, the first six must be non-spades and the seventh must be a spade.

The probability that the first card is not a spade is 39/52 (you have to have chosen one of the 39 non-spades).

This leaves 51 cards with 38 of them non-spades and 13 spades.

The probability that the second card is not a spade is 38/51

This leaves 50 cards with 37 of them non-spades and still 13 spades.

The probability that the third card is not a spade is 37/50

... continue this way until you have drawn six cards.  You now have 46 cards left of which 13 are spades, so the probability of drawing a spade on the seventh draw is 13/46.

You multiply all the above probabilities together to get the overall probability of getting a spade exactly on the seventh draw.

This all assumes you do the drawing without replacing any card after it has been drawn.

.

May 2, 2015

#1
+27902
+5

Assuming the drawing is done without replacement:

1)  probability = 39/52*38/51*37/50*36/49*35/48*34/47*13/46 = 27417/605360 ≈ 0.045

2) We know with certainty that none of the first 5 cards is a spade (because we are told so!).  So probability = 34/47*13/46 = 221/1081 ≈ 0.204

.

May 1, 2015
#2
+12
+5

Thanks Alan but could you please explain in details?

May 1, 2015
#3
+27902
+5

1.  There are 52 cards in a pack.  13 of them are spades and the other 39 are not.

For there to be exactly seven draws before you get a spade, the first six must be non-spades and the seventh must be a spade.

The probability that the first card is not a spade is 39/52 (you have to have chosen one of the 39 non-spades).

This leaves 51 cards with 38 of them non-spades and 13 spades.

The probability that the second card is not a spade is 38/51

This leaves 50 cards with 37 of them non-spades and still 13 spades.

The probability that the third card is not a spade is 37/50

... continue this way until you have drawn six cards.  You now have 46 cards left of which 13 are spades, so the probability of drawing a spade on the seventh draw is 13/46.

You multiply all the above probabilities together to get the overall probability of getting a spade exactly on the seventh draw.

This all assumes you do the drawing without replacing any card after it has been drawn.

.

Alan May 2, 2015