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