Calculating the Probability of Drawing Exactly Two Hearts from Four Cards Using a Standard Deck

When dealing with probabilities involving a deck of cards, the hypergeometric distribution is a powerful tool. This article will explore how to calculate the probability of drawing exactly two hearts from a randomly shuffled standard deck of 52 cards. We'll provide a step-by-step guide and use the hypergeometric distribution to achieve our result.

Overview of Hypergeometric Distribution

The hypergeometric distribution is appropriate in situations where we are drawing items from a finite population without replacement. This distribution is crucial in scenarios like drawing cards from a deck. Let's break down the parameters that we need to calculate the probability:

Total number of cards N 52 Total number of hearts K 13 Total number of cards drawn n 4 Desired number of hearts k 2

Hypergeometric Probability Formula

The probability of drawing exactly k successes (in our case, hearts) in n draws can be calculated using the following formula:

$$P(X k) frac{binom{K}{k} cdot binom{N-K}{n-k}}{binom{N}{n}}$$

Where ({n choose a}) is the binomial coefficient representing the number of ways to choose b successes from a total items.

Step-by-Step Calculation

To make the calculation clearer, let's break down each step:

Step 1: Calculate the Binomial Coefficients

K choose k: ({13 choose 2}) N-K choose n-k: ({39 choose 2}) N choose n: ({52 choose 4})

The binomial coefficients are calculated as follows:

({13 choose 2} frac{13 times 12}{2 times 1} 78) ({39 choose 2} frac{39 times 38}{2 times 1} 741) ({52 choose 4} frac{52 times 51 times 50 times 49}{4 times 3 times 2 times 1} 270725)

Step 2: Plug the Values into the Formula

Now that we have the necessary binomial coefficients, let's plug them into the hypergeometric probability formula:

$$P(X 2) frac{78 cdot 741}{270725} frac{57858}{270725} approx 0.213$$

The final probability of drawing exactly two hearts from four cards is approximately 0.213, or 21.3%.

Interpretation

This result means that if you draw four cards from a well-shuffled standard deck, there is a 21.3% chance that exactly two of those cards will be hearts. It's important to note that probabilities in such scenarios involve a large number of possible outcomes, each with varying levels of likelihood.

Summary

The hypergeometric distribution is an essential tool for calculating probabilities in situations with finite, distinct, and without replacement items, such as drawing cards from a deck. By understanding and applying this method, you can effectively determine the probabilities of various outcomes, enhancing your ability to make informed decisions in card games and other probabilistic scenarios.