Probability question

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\(\large \color{black}{\begin{align}& \normalsize \text{Find the probability that when a hand of 7 cards is drawn from a well}\hspace{.33em}\\~\\ & \normalsize \text{shuffled deck of 52 cards, it contains atleast 3 Kings.}\hspace{.33em}\\~\\ \end{align}}\)
Total number of possible hands = 52C7
How many kings are there in a full deck ?

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Other answers:

4
  • 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
what u mean ?
  • phi
how many ways can you choose 4 "not kings" from the deck?
48
  • phi
48C4
yes
  • 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
  • phi
next, we want the number of hands with 4 kings
4C3\(\times \)48C4
4C4×48C3
  • phi
ok, now add that is the numerator (# of hands that match our criteria) as you know the denominator is 52C7
=\(\dfrac{4C4×48C3+4C3×48C4}{52C7}\)
  • phi
yes
thnks
  • phi
yw
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.

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