The deck of a card game is made up of 108 cards. 25 each are red, yellow, blue, and green, and 8 are wild cards. Each player is randomly dealt a 7-card hand. What is the probability that a hand will contain (a) exactly 2 wild cards, and (b) 2 wild cards, 2 red cards, and 3 blue cards?
Please explain how to do this.
Please explain how to do this.
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There are (108 choose 7) possible 7-card hands from 108 cards.
(a) There are (8 choose 2) ways of choosing 2 wild cards from the 8 available.
There are (100 choose 5) ways of choosing 5 non-wild cards from the 100 available.
Therefore, by the fundamental counting principle, there are
(8 choose 2)*(100 choose 5) possible hands with exactly 2 wild cards.
So P(exactly 2 wild cards) = (8 choose 2)*(100 choose 5) / (108 choose 7)
= 0.0756 from using a TI-89 calculator.
(b) There are (8 choose 2) ways of choosing 2 wild cards from the 8 available.
There are (25 choose 2) ways of choosing 2 red cards from the 25 available.
There are (25 choose 3) ways of choosing 3 blue cards from the 25 available.
Note that 2 + 2 + 3 = 7, so the entire 7-card hand has been considered.
Therefore, by the fundamental counting principle, there are
(8 choose 2)*(25 choose 2)*(25 choose 3) possible hands with 2 wild cards, 2 red cards, and 3 blue cards.
So P(2 wild cards, 2 red cards, and 3 blue cards)
= (8 choose 2)*(25 choose 2)*(25 choose 3) / (108 choose 7)
= 0.000693 from using a TI-89 calculator.
Note: the first answerer should have multiplied the result in (a) by (7 choose 2), i.e. the number of ways of choosing which 2 of the 7 draws are the wild cards, and should have multiplied the result in (b) by the product (7 choose 2)*(5 choose 2), i.e. the number of ways of choosing which 2 of the 7 draws are the wild cards and then choosing which 2 of the 5 non-wild card draws are red cards.
Lord bless you today!
(a) There are (8 choose 2) ways of choosing 2 wild cards from the 8 available.
There are (100 choose 5) ways of choosing 5 non-wild cards from the 100 available.
Therefore, by the fundamental counting principle, there are
(8 choose 2)*(100 choose 5) possible hands with exactly 2 wild cards.
So P(exactly 2 wild cards) = (8 choose 2)*(100 choose 5) / (108 choose 7)
= 0.0756 from using a TI-89 calculator.
(b) There are (8 choose 2) ways of choosing 2 wild cards from the 8 available.
There are (25 choose 2) ways of choosing 2 red cards from the 25 available.
There are (25 choose 3) ways of choosing 3 blue cards from the 25 available.
Note that 2 + 2 + 3 = 7, so the entire 7-card hand has been considered.
Therefore, by the fundamental counting principle, there are
(8 choose 2)*(25 choose 2)*(25 choose 3) possible hands with 2 wild cards, 2 red cards, and 3 blue cards.
So P(2 wild cards, 2 red cards, and 3 blue cards)
= (8 choose 2)*(25 choose 2)*(25 choose 3) / (108 choose 7)
= 0.000693 from using a TI-89 calculator.
Note: the first answerer should have multiplied the result in (a) by (7 choose 2), i.e. the number of ways of choosing which 2 of the 7 draws are the wild cards, and should have multiplied the result in (b) by the product (7 choose 2)*(5 choose 2), i.e. the number of ways of choosing which 2 of the 7 draws are the wild cards and then choosing which 2 of the 5 non-wild card draws are red cards.
Lord bless you today!
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you have 8/108 for 1st wild card, 7/107 for second wild card. 100/106 for 3rd card, 99/105 for 4th card , etc etc and 96/102 for 8th card. total probability is product of all this
a) (8/108) (7/107)(100 /106)(99/105)(98/104)(97/103)(96/102) = 0.0036
b) you have 8/108 for 1st wild card, 7/107 for second wild card. 25/106 for 1 red card, 24/105 for 2nd red card , etc etc and 23/102 for 3rd blue card. total probability is product of all this
(8/108)(7/107)(25/106)(24/105) (25/104) (24/103) (23/102)
= 0.00000329
a) (8/108) (7/107)(100 /106)(99/105)(98/104)(97/103)(96/102) = 0.0036
b) you have 8/108 for 1st wild card, 7/107 for second wild card. 25/106 for 1 red card, 24/105 for 2nd red card , etc etc and 23/102 for 3rd blue card. total probability is product of all this
(8/108)(7/107)(25/106)(24/105) (25/104) (24/103) (23/102)
= 0.00000329