Question

5. A royal flush consists of an ace, king, queen, jack, and 10 all in the same suit. If 7 cards are dealt at random from a standard deck, determine the probability of getting a) a royal flush in spades. b) a royal flush in any suit.

Answer 1

Please help!

Answer 2

So first how many cards are in a deck?

Answer 3

@iowastate

Answer 4

52

Answer 5

okay now how many different suits are there?

Answer 6

4

Answer 7

there are a couple of ways to do this
you have 5 cards to choose and 52 total
probability the first card is one of those 5 is \(\frac{5}{52}\)
now there are 4 cards to choose from and 51 cards, next probability is \(\frac{4}{51}\) and so on
multiply to get your answer

Answer 8

There are \({52 \choose 7}\) possible ways to deal the card.
You want

1) Ways of grouping 5 cards that make the royal flush in a single group of 5 times ways of grouping the remaining 47 cards in a group of two:\({5\choose 5}\times {47\choose2}\)
2) Number of suits that will form a royal flush:\(4\choose 1\)
3) Possible ways to deal cards:\({52 \choose 7}\)

But you are looking only for a royal flush that is a \(spade\), which is just 1/4 of the previous result because there are 4 suits and thus four ways to win.

$$
\large
\cfrac{1}{4}\cfrac{{4\choose 1}{5\choose 5}{47\choose2}}{{52\choose 7}}
$$