+12 Please help,
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?
+33614 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.
.
+33614 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
.
+33614 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.
.
