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 1,287 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 63,206 of the possible 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?

Short answer

The probabilities are roughly 0.000495 for a hand consisting entirely of spades, 0.00198 for a hand consisting entirely of a single suit and 0.02432 for a hand consisting entirely of spades and clubs with both suits represented.

Step-by-step solution

Compute the probability of a hand consisting entirely of spades

The total number of possible hands is 2,598,960. Among these, there are 1,287 hands that consist entirely of spades. The probability is thus calculated by dividing the number of desirable outcomes (entirely spade hands) by the total number of outcomes, which results in \( \frac{1287}{2,598,960} \).

Compute the probability of a hand consisting entirely of a single suit

A standard deck has 4 suits, so the possible hands that consist entirely of a single suit are 4 times the number of pure spade hands. Thus, the probability can be calculated as \( \frac{4 \times 1287}{2,598,960} \).

Compute the probability of a hand consists entirely of spades and clubs with both suits represented

If we sum the number of hands that only consist of spades and clubs, we get 63,206. Again, we divide this by the total number of outcomes to get the probability, which is \( \frac{63,206}{2,598,960} \).

Key concepts

Probability Theory

Probability theory is a branch of mathematics that deals with quantifying the likelihood of events. It is fundamental in understanding how often we can expect certain outcomes to occur during random processes, such as dealing cards from a shuffled deck. In card games, each card drawn alters the composition of the deck and therefore changes the probabilities of future events.

For example, in the exercise provided, the concept of probability is applied to determine how likely it is to draw a hand consisting entirely of spades, or one consisting of just spades and clubs. The calculations hinge on a key principle of probability: the ratio of the number of favorable outcomes to the total number of possible outcomes.

In a 52-card deck, there are initially an equal number of cards in each suit, so the probability of drawing a card from any particular suit is the same. As we draw more cards without replacement, the probability of drawing a card from a specific suit changes in a predictable way, which can be mathematically analyzed using probability theory.

Combinatorics in Probability

Combinatorics is a field of mathematics primarily concerned with counting, combinations, and permutations, which is crucial when calculating probabilities, especially in games like poker where the order of events can significantly impact outcomes. The exercise demonstrates combinatorics through the counting of possible hands from a standard deck.

A standard 52-card deck can be arranged in a vast number of possible combinations when dealt into a hand of five cards. This is often calculated using the combinatorial function known as 'n choose k', which is the number of ways to choose k items from a set of n, regardless of order. When you multiply the number of combinations by the number of suits, as in Step 2 of the solution, you use combinatorial principles to find the number of hands consisting of a single suit.

Combinatorics is particularly helpful for understanding the large quantities involved in card probabilities and can simplify seemingly complex problems into more manageable calculations.

Statistical Outcomes

In the context of probability in card games, 'statistical outcomes' refers to the results that arise from the multitude of ways cards can be distributed. These outcomes are fundamental to making predictions about future events, such as the likelihood of being dealt a particular hand.

In the exercise, statistical outcomes are highlighted by the number of possible hands that can be dealt from the deck. With 2,598,960 total hands possible, grasping the scale and using it to calculate probabilities is a key part of solving probability problems. These outcomes are also influenced by conditions, such as requiring that a hand consists of only spades and clubs with both suits represented, which reduces the number of favorable outcomes to 63,206 for that specific scenario.

Understanding how to work with statistical outcomes is pivotal when applying probability theory to real-life situations. It allows us to make informed decisions based on the likelihood of various events, which is valuable in fields ranging from gaming to finance.