Question

You pick three cards at random from a deck. Find the probability of each event described below. a) You get no aces. b) You get all hearts. c) The third card is your first red card. d) You have at least one diamond.

Short answer

a) ≈ 0.7829, b) ≈ 0.0129, c) ≈ 0.3824, d) ≈ 0.5867.

Step-by-step solution

Calculate Total Possible Outcomes

First, calculate the total number of ways to pick 3 cards from a deck of 52 cards. The formula for combinations is \( nCr = \frac{n!}{r!(n-r)!} \). Here, \( n = 52 \) and \( r = 3 \).\[ \binom{52}{3} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = 22,100 \]This is the total number of possible outcomes when drawing 3 cards.

Find Probability of No Aces

There are 48 non-ace cards (since there are 4 aces in a deck). The number of ways to select 3 non-ace cards is \( \binom{48}{3} \).\[ \binom{48}{3} = \frac{48 \times 47 \times 46}{3 \times 2 \times 1} = 17,296 \]The probability of getting no aces is then:\[ P(\text{no aces}) = \frac{\binom{48}{3}}{\binom{52}{3}} = \frac{17,296}{22,100} \approx 0.7829 \]

Find Probability of All Hearts

There are 13 hearts in a deck. The number of ways to select 3 heart cards is \( \binom{13}{3} \).\[ \binom{13}{3} = \frac{13 \times 12 \times 11}{3 \times 2 \times 1} = 286 \]The probability of getting all hearts is:\[ P(\text{all hearts}) = \frac{\binom{13}{3}}{\binom{52}{3}} = \frac{286}{22,100} \approx 0.0129 \]

Find Probability Third Card is First Red

To find this probability, consider cases where the first two cards drawn are black (non-red) and the third is red.1. Select 2 black cards: There are 26 black cards, so the number of ways to pick 2 black cards is \( \binom{26}{2} = 325 \).2. Select 1 red card as the third card out of 26 red cards: 26 ways.Total favorable outcomes: \( 325 \times 26 = 8,450 \).The probability is:\[ P(\text{third card first red}) = \frac{8,450}{22,100} \approx 0.3824 \]

Find Probability of at Least One Diamond

Use the complement rule. First, calculate the probability of getting no diamonds.1. There are 39 non-diamond cards. 2. Number of ways to select 3 non-diamond cards: \( \binom{39}{3} = 9,139 \).Probability of no diamonds:\[ P(\text{no diamonds}) = \frac{9,139}{22,100} \approx 0.4133 \]Probability of at least one diamond:\[ P(\text{at least one diamond}) = 1 - P(\text{no diamonds}) = 1 - 0.4133 = 0.5867 \]

Key concepts

Combinations

Combinations provide a way to determine how many different ways you can choose a specific number of items from a larger group. In probability, combinations are important because they help us calculate the total number of possible outcomes.
When we talk about combinations, we usually denote them using the formula:

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

where \( n \) is the total number of items, \( r \) is the number of items to choose, and \(!\) signifies "factorial," which means multiplying a sequence of descending natural numbers. For example, \( 3! = 3 \times 2 \times 1 = 6 \). This formula tells us the number of different ways to choose \( r \) items from \( n \) without regard to order. Hence, it's perfect for cases where the order of selection does not matter.
In our problem, to find how many ways you can draw three cards from a full deck, you use \( \binom{52}{3} \). This combination gives us all possible three-card draws regardless of the order. Combinations, therefore, offer us the foundational numbers we need to calculate probabilities in more complex scenarios.

Deck of Cards

A standard deck of cards is a common tool used in probability examples due to its well-defined structure and familiar configuration. Understanding the deck's makeup is crucial for calculating probabilities in card-related problems.
A typical deck consists of 52 cards, divided into four suits:

• Hearts
• Diamonds
• Clubs
• Spades

Each suit contains 13 cards, numbered from Ace through ten, and includes three face cards: Jack, Queen, and King.
These suits are further grouped into two colors: red (hearts and diamonds) and black (clubs and spades), each containing 26 cards. This even distribution is helpful when calculating the probability of drawing certain suits or colors. For example, drawing a card that is from the hearts suit or finding the likelihood that a certain card is not an ace.
By understanding the structure of a deck, you can break down problems into manageable pieces using combinations and other probability strategies.

Complement Rule

The complement rule in probability is a technique used to simplify the calculation of probabilities of certain events. Sometimes, it's more straightforward to calculate the probability of an event not happening and then use the complement to find the probability of the event happening.
The complement rule states:

• \( P(A) = 1 - P(A^c) \)

where \( P(A) \) is the probability of event \( A \) occurring, while \( P(A^c) \) is the probability of event \( A \) not occurring.
In scenarios involving "at least" one occurrence, like finding the probability of drawing at least one diamond from three cards, calculating the complement (drawing no diamonds) first can simplify the problem.
For instance, rather than directly calculating the cases where at least one card is a diamond, you determine the probability of drawing zero diamonds and subtract this from 1. This is often much easier because not drawing a diamond involves a smaller, simpler set of calculations. Employing the complement rule is a potent tool, especially in probability exercises involving exhaustive or complicated outcome sets.