## mathmath333 one year ago Probability question

1. mathmath333

\large \color{black}{\begin{align}& \normalsize \text{Find the probability that when a hand of 7 cards is drawn from a well}\hspace{.33em}\\~\\ & \normalsize \text{shuffled deck of 52 cards, it contains atleast 3 Kings.}\hspace{.33em}\\~\\ \end{align}}

2. mathmath333

Total number of possible hands = 52C7

3. ganeshie8

How many kings are there in a full deck ?

4. mathmath333

4

5. phi

break the "good" hands into the problem of choosing 4 "not kings" and 3 kings plus prob of choosing 3 not kings, and 4 kings

6. mathmath333

what u mean ?

7. phi

how many ways can you choose 4 "not kings" from the deck?

8. mathmath333

48

9. phi

48C4

10. mathmath333

yes

11. phi

that gives you all the "4 card" hands with no kings now pick 3 kings from the king pile (4 cards) 4C3 multiply to get the number of hands with exactly 3 kings

12. phi

next, we want the number of hands with 4 kings

13. mathmath333

4C3$$\times$$48C4

14. mathmath333

4C4×48C3

15. phi

ok, now add that is the numerator (# of hands that match our criteria) as you know the denominator is 52C7

16. mathmath333

=$$\dfrac{4C4×48C3+4C3×48C4}{52C7}$$

17. phi

yes

18. mathmath333

thnks

19. phi

yw

20. Loser66

Thank you very much @phi . I am always in trouble with probability. Your method is PERFECT. It is easy to understand what is going on. Thanks you so much.