Question

Four cards are dealt at random from a deck. What is the probability that at least one of them is an Ace? The answer may be given in terms of the combinatorial notation \(C(a, b)\).

Short answer

The probability is \(1 - \frac{C(48, 4)}{C(52, 4)}\).

Step-by-step solution

Understand the Scenario

We are asked to find the probability that at least one of the four cards dealt from a deck is an Ace. There are 52 cards in a standard deck, including 4 Aces.

Calculate Total Possible Outcomes

The total number of ways to deal 4 cards from a deck of 52 cards is given by the combination formula: \[ C(52, 4) = \frac{52!}{4!(52-4)!} \]

Calculate the Complement Probability

To find the probability of at least one Ace, it is easier to first find the probability of getting no Aces at all. There are 48 non-Ace cards in a deck. The number of ways to choose 4 cards from these 48 is:\[ C(48, 4) = \frac{48!}{4!(48-4)!} \]

Calculate the Probability of No Aces

The probability of getting no Aces (only dealing non-Ace cards) is:\[ \frac{C(48, 4)}{C(52, 4)} \]

Calculate the Probability of At Least One Ace

The probability of getting at least one Ace is the complement of the previous step:\[ 1 - \frac{C(48, 4)}{C(52, 4)} \]

Express the Final Probability

The probability that at least one of the dealt cards is an Ace is:\[ 1 - \frac{C(48, 4)}{C(52, 4)} \]

Key concepts

Combinatorics

Combinatorics is a fascinating area of mathematics that deals with the counting, arrangement, and combination of objects. One of the core concepts in combinatorics is the idea of combinations, which are selections of items where the order does not matter.

In the context of drawing cards from a deck, combinations can be used to determine how many different ways we can select a certain number of cards. For example, to calculate how many ways 4 cards can be selected from a standard deck of 52, we use the combination formula:

• It is represented as \( C(n, k) \) or sometimes \( \binom{n}{k} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.
• The formula is \( C(n, k) = \frac{n!}{k!(n-k)!} \).

In our card example, it means we have \( C(52, 4) = \frac{52!}{4!(52-4)!} \) ways to draw 4 cards from a deck without considering the order of cards. Combinatorics provides the tools necessary to solve such problems and is essential for calculating probabilities in various scenarios.

Complement Rule

The complement rule is a handy principle in probability that simplifies complex problems. It states that the probability of an event occurring is one minus the probability of it not occurring.

This rule is especially useful when calculating probabilities of complex events like 'at least one' scenarios. Instead of calculating the probability of 'at least one Ace,' which involves multiple cases, we find it easier to calculate the probability of 'no Aces' and subtract it from 1.

• Consider the probability of at least one Ace when 4 cards are drawn.
• The complement event (no Aces) is finding the probability of drawing 4 cards, none of which are Aces.
• Calculate \( C(48, 4) \) since there are 48 cards that are not Aces.

Thus, the probability of 'at least one Ace' equals \( 1 - \frac{C(48, 4)}{C(52, 4)} \). This approach simplifies the calculation considerably.

Combinatorial Probability

Combinatorial probability involves calculating the likelihood of an event based on combinatorial principles. It effectively combines the power of counting methods from combinatorics with probability theory.

When we compute probabilities using combinations, we're dealing with combinatorial probability. This method is particularly suitable when we're interested in scenarios involving random selection from a group.

To find the probability that, for example, at least one card is an Ace from 4 cards drawn, we use:

• First, compute the total number of ways to draw the 4 cards: \( C(52, 4) \).
• Then calculate the number of ways to draw 4 non-Ace cards: \( C(48, 4) \).
• The probability of no Aces is \( \frac{C(48, 4)}{C(52, 4)} \).
• Finally, use the complement rule to find the probability that at least one card is an Ace: \( 1 - \frac{C(48, 4)}{C(52, 4)} \).

By leveraging combinatorial probability, we gain insight into the structured nature of probabilities in situations of random selection, making complex calculations more manageable.