Question

These problems have to do with combining probability. Surprisingly, I am struggling with them. For more clarification,the questions ask "what is the probability?"

From The Combining Probabilities Handout 2. (16) Drawing three hearts in a row from a regular deck of cards when the drawn card is not returned to the deck each time. 3.120) Being dealt four red cards off the top of a regular deck of well-shuffled cards. 4. (22) Randomly selecting a three-person committee consisting entirely of Americans froma pool of 12 British people and 18 Americans.

Answer 1

2.

We need to find P[ drawing 3 hearts in a row from a regular deck of cards where the drawn card is not returned ]

Number of hearts in a regular deck of cards = 13

Total number of cards in a regular deck of cards = 52

P[ drawing a hearts in first trial ] = Number of hearts in a regular deck of cards / Total number of cards in a regular deck of cards

P[ drawing a hearts in first trial ] = 13/52 = 1/4

P[ drawing a hearts in first trial ] = 1/4

After one draw,

Number of hearts in the deck of cards after first draw = 12

Total number of cards in the deck of cards after first draw = 51

P[ drawing a hearts in second trial ] = Number of hearts in the deck of cards after first draw / Total number of cards in the deck of cards after first draw = 12/51 = 4/17

P[ drawing two successive hearts in two trials ] = P[ drawing a hearts in first trial ]*P[ drawing a hearts in second trial ]

P[ drawing two successive hearts in two trials ] = (1/4)*(4/17) = 1/17

P[ drawing two successive hearts in two trials ] = 1/17

After two draw,

Number of hearts in the deck of cards after second draw = 11

Total number of cards in the deck of cards after second draw = 50

P[ drawing a hearts in third trial ] = Number of hearts in the deck of cards after second draw / Total number of cards in the deck of cards after second draw = 11/50

P[ drawing three successive hearts in three trials ] = P[ drawing a hearts in two successive trial ]*P[ drawing a hearts in third trial ]

P[ drawing three successive hearts in three trials ] = (1/17)*(11/50) = 11/850

P[ drawing three successive hearts in three trials ] = 0.0129

Alternate,

Drawing 3 cards out of 13 from a deck of 52 cards.

P[ drawing three successive hearts in three trials ] = drawing 3 cards from 13 hearts / drawing 3 cards from 52 cards

P[ drawing three successive hearts in three trials ] = 13C3 / 52C3

P[ drawing three successive hearts in three trials ] = 13!/(3!*(13-3)!) / 52!/(3!*(52-3)!)

P[ drawing three successive hearts in three trials ] = 13!/(3!*10!)/52!/(3!*49!)

P[ drawing three successive hearts in three trials ] = (13*12*11/3*2*1)/(52*51*50/3*2*1)

P[ drawing three successive hearts in three trials ] = 286/22100

P[ drawing three successive hearts in three trials ] = 0.0129

3.

Number of red cards in a regular deck = 26

Total number of cards in a regular deck = 52

Being 4 cards of red color the top on a deck of 52 cards.

P[ Being 4 cards of red color the top ] = 4 cards from 26 red cards / 4 cards from 52 cards

P[ Being 4 cards of red color the top ] = 26C4 / 52C4

P[ Being 4 cards of red color the top ] = 26!/(4!*(26-4)!) / 52!/(4!*(52-4)!)

P[ Being 4 cards of red color the top ] = 26!/(4!*22!)/52!/(4!*48!)

P[ Being 4 cards of red color the top ] = (26*25*24*23/4*3*2*1)/(52*51*50*49/43*2*1)

P[ Being 4 cards of red color the top ] = 14950/270725

P[ Being 4 cards of red color the top ] = 0.0552

4.

12 British and 18 Americans

Total number of members = 12 + 18 = 30

P[ Selecting all 3 Americans in the committee ] = Selecting 3 Americans of out 18 / Selecting 3 people out of 30

P[ Selecting all 3 Americans in the committee ] = 18C3 / 30C3

P[ Selecting all 3 Americans in the committee ] = 18!/(3!*(18-3)!) / 30!/(3!/(30-3)!)

P[ Selecting all 3 Americans in the committee ] = 18!/(3!*15!) / 30!/(3!*27!)

P[ Selecting all 3 Americans in the committee ] = (18*17*16/3*2*1)/(30*29*28/3*2*1)

P[ Selecting all 3 Americans in the committee ] = 816/4060

P[ Selecting all 3 Americans in the committee ] = 0.2009