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Chapter 11: Problem 11

Drawing Cards. Suppose that 3 cards are drawn from a well-shuffled deck of 52 cards. What is the probability that they are all aces?

Short Answer

Expert verified
The probability of drawing 3 aces is approximately 0.000181.

Step by step solution

01

Determine the total number of ways to draw 3 cards

Calculate the total number of combinations of drawing 3 cards from a deck of 52. This can be found using the combination formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \( n = 52 \) and \( k = 3 \).Therefore, \[ \binom{52}{3} = \frac{52!}{3!(52-3)!} = \frac{52!}{3!\cdot49!} \].
02

Simplify the total number of combinations

Simplify the expression: \[ \binom{52}{3} = \frac{52 \times 51 \times 50}{3 \times 2 \times 1} = 22100 \].So, the total number of ways to draw 3 cards from 52 is 22,100.
03

Determine the number of ways to draw 3 aces

Calculate the number of combinations to draw 3 aces from the 4 aces available using the same combination formula. Here, \( n = 4 \) and \( k = 3 \).\[ \binom{4}{3} = \frac{4!}{3!(4-3)!} = \frac{4!}{3!\cdot1!} \].
04

Simplify the number of combinations for aces

Simplify the expression: \[ \binom{4}{3} = \frac{4 \times 3!}{3! \times 1} = 4 \].So, there are 4 ways to draw 3 aces from 4 aces.
05

Calculate the probability

Calculate the probability of drawing 3 aces by dividing the number of ways to draw 3 aces (4) by the total number of ways to draw 3 cards (22,100):\[ P = \frac{\text{Number of ways to draw 3 aces}}{\text{Total number of ways to draw 3 cards}} = \frac{4}{22100} \approx 0.000181 \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

combinations
Combinations are a fundamental concept in mathematics used to determine the number of ways to select items from a larger set, without regard to the order of selection. For example, if you want to know how many ways you can choose 3 cards from a 52-card deck, you use combinations. This is particularly useful in probability and statistics. To find combinations, we use the formula: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items and \( k \) is the number of items to choose. This formula takes into account that the order of choosing does not matter.
binomial coefficient
The binomial coefficient, often represented as \( \binom{n}{k} \), is a specific type of combination that counts the number of ways to choose \( k \) items from \( n \) items. It's the same formula mentioned above, and it plays a crucial role in binomial expansions and various probability problems. For example, in the given exercise, to compute the number of ways to choose 3 cards out of 52, we use \( \binom{52}{3} = \frac{52!}{3!(52-3)!} \). This calculation tells us that there are 22,100 possible combinations of drawing 3 cards from a standard deck of 52 cards.
probability calculation
Probability calculation involves determining the chance of a particular event occurring. It's calculated by dividing the number of successful outcomes by the total number of possible outcomes. In our case, the probability of drawing 3 aces from a deck of 52 cards is calculated as follows: First, we find the number of favorable outcomes (ways to draw 3 aces), then divide it by the total number of ways to draw any 3 cards from the deck. Using our numbers, the probability is \( P = \frac{\text{Number of ways to draw 3 aces}}{\text{Total number of ways to draw 3 cards}} = \frac{4}{22100} \). This results in a very small probability, approximately 0.000181.
factorial
Factorials play an important role in combinations and probability calculations. A factorial, denoted by an exclamation point (!), is the product of all positive integers up to a given number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). In the combinations formula \( \binom{n}{k} \), factorials are used to calculate how many ways we can choose \( k \) items from \( n \) items. Simplifying the factorial in our problem helps us reduce complex calculations, like in \( \binom{52}{3} = \frac{52!}{3!(52-3)!} = \frac{52 \times 51 \times 50}{3!} \), making it easier to arrive at the result.

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