3. Find the number of ways to draw a five-card hand consisting of. (a) Any five

Question

3. Find the number of ways to draw a five-card hand consisting of. (a) Any five cards. (b) All Spades. (c) Any five cards of the same suit. (d) Three jacks and two aces. (e) Three face cards. 4. Suppose that in a cooler is 24 sodas: 8 Root Beer, 8 Orange and 8 Cola. If you pull out five sodas from the cooler at random, then find the probability that: (a) All five are Root Beer. b) All five are either Orange or Cola. (c) All five are one kind. The kind is not specified.) 5. Find the probability of drawing a seven-card hand with.. (a) No aces (b) Exactly one ace. (c) Exactly two aces (d) At least one ace. Hint: Pat least one ace)- 1-P(mn)

Explanation / Answer

3. Five card hand :

Number of cards in the deck =52;

(a) Any five cards ;

Number of ways drawing any 5 cards from deck of 52 cards is: 52C5 = (52x51x50x49x48) / (1x2x3x4x5) = 2598960

(b) All Spades ;

Number of spade cards in the deck = 13

Number of ways drawing any 5 cards from 13 spade cards: 13C5 = (13x12x11x10x9) / (1x2x3x4x5) = 1287

(c) Any five cards of the same suit.

Number of suits =4 ;

Spade suit ;

Number of ways drawing any 5 cards from 13 spade cards: 13C5 = (13x12x11x10x9) / (1x2x3x4x5) = 1287

Similarly,

Hearts suit ;

Number of ways drawing any 5 cards from 13 hearts cards: 13C5 = (13x12x11x10x9) / (1x2x3x4x5) = 1287

Diamonds suit ;

Number of ways drawing any 5 cards from 13 Diamond cards: 13C5 = (13x12x11x10x9) / (1x2x3x4x5) = 1287

Clubs suit :

Number of ways drawing any 5 cards from 13 Clubs cards: 13C5 = (13x12x11x10x9) / (1x2x3x4x5) = 1287

Number of ways of drawing five cards from any suit =

Number of ways of drawing five cards from spades + Number of ways of drawing five cards from Hearts + Number of ways of drawing five cards from Diamonds or Number of ways of drawing five cards from Clubs

= 1287 + 1287 +1287 +1287 = 5148

d. Three jacks and two aces

Number of jacks in the deck = 4

Number of aces in the deck = 4

Number of ways of drawing three jacks from four jacks = 4C3 = (4x3x2) / (1X2x3) = 24/6 = 4

Number of ways of drawing 2 aces from 4 aces = 4C2 = (4x3) / (1X2) = 12/2 = 6

Number of ways of drawing 3 jacks and two aces = 4 x 6 =24

e. Three face cards :

Number faces cards = 12 (4 jack, 4 queens, 4 kings)

Total number of cards to be drawn = 5

Number of ways of drawing 3 face cards from 12 face cards = 12C3 = (12x11x10) / (1x2x3) = 1320 / 6 = 220

Number of ways drawing remaining 2 cards( 5-3) from the remainng (52-12) 40 cards = 40C2 = (40X39) / (1x2) = 780

Number of ways drawing 3 face cards = 220 x 780 = 171600

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