Question

A deck of 52 cards is mixed well, and 5 cards are dealt. a. It can be shown that (disregarding the order in which the cards are dealt) there are \(2,598,960\) possible hands, of which only 1287 are hands consisting entirely of spades. What is the probability that a hand will consist entirely of spades? What is the probability that a hand will consist entirely of a single suit? b. It can be shown that exactly 63,206 hands contain only spades and clubs, with both suits represented. What is the probability that a hand consists entirely of spades and clubs with both suits represented? c. Using the result of Part (b), what is the probability that a hand contains cards from exactly two suits?

Short answer

The probabilities are as follows: a. The probability that a hand will consist entirely of spades is \(\frac{1287}{2598960}\). The probability that a hand will consist entirely of a single suit is \(4 * \frac{1287}{2598960}\). b. The probability that a hand consists entirely of spades and clubs with both suits represented is \(\frac{63206}{2598960}\). c. The probability that a hand contains cards from exactly two suits is \(6 * \frac{63206}{2598960}\).

Step-by-step solution

Probability of a hand consisting entirely of spades

The probability is calculated as the ratio of the number of favorable outcomes to the total number of outcomes. Here, it's known that there are 1,287 hands consisting entirely of spades out of a total of 2,598,960 hands. So, the probability is calculated as \(\frac{1287}{2598960}\)

Probability of a hand consisting entirely of a single suit

Remember that a deck of cards consists of 4 different suits: spades, hearts, diamonds, and clubs. So the probability of getting a hand consisting entirely of any single suit would be four times the probability of getting a hand consisting entirely of spades. So, the probability would be \(4 * \frac{1287}{2598960}\)

Probability of a hand consisting entirely of spades and clubs

Again, the probability is calculated as the ratio of the number of favorable outcomes to the total number of outcomes. Here, it's known that there are 63,206 hands consisting entirely of spades and clubs (with both suits represented) out of a total of 2,598,960 hands. So, the probability is calculated as \(\frac{63206}{2598960}\)

Probability of a hand containing cards from exactly two suits

A standard deck has 4 different suits, so there are \(\binom{4}{2} = 6\) ways to select 2 suits. Now from the results in Step 3, each has the probability \(\frac{63206}{2598960}\). So, the probability of a hand containing cards from exactly two suits would be \(6 * \frac{63206}{2598960}\)