Question

find the probability taht when a hand of 7 cards is drawn from a well shufflrd deck of 52 cards it contains 1) 3 kings .............2) atleast 3 kings​

Answer 1

Answer:

1) 5.81771 x $10^{-3}$ .

2) 5.94699 x $10^{-3}$

Step-by-step explanation:

Since we know that in a well shuffled deck of 52 cards there are only 4 kings.

1) Probability that when a hand of 7 cards is drawn from a well shuffled deck of 52 cards it will contain 3 kings is given by ;

 3 kings is drawn out of total 4 kings and remaining 4 cards are drawn from 48 cards

Number of Combination of choosing 3 kings out of total 4 kings = $^{4}C_3$  

And choosing remaining 4 cards are drawn from 48 cards = $^{48}C_4$

Total combinations of choosing 7 cards from total of 52 cards = $^{52}C_7$

Hence, required probability = $\frac{^{4}C_3 \times ^{48}C_4}{^{52}C_7}$ = $\frac{\frac{4!}{3!\times 1!}\times \frac{48!}{4!\times 44!}}{\frac{52!}{7!\times 45!}}$  {Because $^{n}C_r$ = $\frac{n!}{r! * (n-r)!}$ }

                                               = 5.81771 x $10^{-3}$ .

2) Probability that when a hand of 7 cards is drawn from a well shuffled deck of 52 cards it contains at least 3 kings = it contain 3 kings + it contains

                                                                                                                4 kings​

The probability of containing 3 kings has been calculated in first part

So probability of containing 4 kings is given by :

No. of combinations of choosing 4 kings out of total 4 kings = $^{4}C_4$

And choosing remaining 3 cards from remaining 48 cards = $^{48}C_3$

Total combinations of choosing 7 cards from total of 52 cards = $^{52}C_7$

So, the probability of containing 4 kings =  $\frac{^{4}C_4 \times ^{48}C_3}{^{52}C_7}$ = $\frac{\frac{4!}{4!\times 0!}\times \frac{48!}{3!\times 45!}}{\frac{52!}{7!\times 45!}}$

                                                                                     =  1.29282 x $10^{-4}$

Hence, Required probability = 5.81771 x $10^{-3}$ + 1.29282 x $10^{-4}$

                                                = 5.94699 x $10^{-3}$ .

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