I know how to do the probability when I pick 3 card from the deck and all three are kings. But with this question, I am confused because I do not know if the 3 kings are two red and a black or 2 black and one red. So the number of red and black left in the deck after 3 kings are pick out are different (ie 24 reds left if 2 kings are red or 25 reds if 1 king is red).
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$\begingroup$ "3 of 5 are kings" and "the first three are kings" are different events. $\endgroup$– Graham KempCommented Oct 24, 2023 at 3:53
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$\begingroup$ @GrahamKemp Who mentioned picking the cards in a particular order? $\endgroup$– Ted ShifrinCommented Oct 24, 2023 at 4:13
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$\begingroup$ @TedShifrin Yanni Zh in the title. $\endgroup$– Graham KempCommented Oct 24, 2023 at 4:29
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$\begingroup$ @GrahamKemp I do not interpret it that way. It is specifying what cards are drawn. $\endgroup$– Ted ShifrinCommented Oct 24, 2023 at 14:17
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1 Answer
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"$3$ among $5$ are kings" and "the first three are kings" are quite different events.
You have four disjoint events to consider:
- A red non-king in 4th place, a black non-king in 5th place, then $3$ kings among the first three cards.
- A red king in 4th place, a black non-king in 5th place, then $2$ kings and $1$ non-king among the first three cards.
- A red non-king in 4th place, a black king in 5th place, then $2$ kings and $1$ non-king among the first three cards.
- A red king in 4th place, a black king in 5th place, then $1$ king and $2$ non-kings among the first three cards.
answered Oct 24, 2023 at 4:10