Probability of Drawing Two Cards of the Same Suit from a Well-Shuffled Deck
The probability of drawing two cards of the same suit from a well-shuffled deck of 52 standard playing cards is a classic problem in probability theory. This article will explore the method to find this probability using both straightforward and conditional probability approaches, highlighting the importance of combinations in each step.
Understanding the Problem
When two cards are drawn without replacement from a well-shuffled deck, what is the probability that they are of the same suit? To solve this, we will break down the problem into several steps, ensuring clarity and precision in each.
Step 1: Total Number of Ways to Draw 2 Cards from 52 Cards
The first step is to determine the total number of ways to draw 2 cards from a deck of 52 cards, denoted by the combination formula binom{52}{2}.
binom{52}{2} frac{52 times 51}{2} 1326
Step 2: Number of Ways to Draw 2 Cards of the Same Suit
A standard deck has 4 suits, each containing 13 cards. The number of ways to choose 2 cards from a single suit is given by binom{13}{2}.
binom{13}{2} frac{13 times 12}{2} 78
Since there are 4 suits, the total number of ways to draw 2 cards of the same suit is:
4 times 78 312
Step 3: Calculating the Probability
The probability of drawing two cards of the same suit is the ratio of the number of favorable outcomes to the total number of outcomes.
Ptext{same suit} frac{312}{1326}
This fraction can be simplified using the greatest common divisor (GCD) of 312 and 1326, which is 78.
frac{312 div 78}{1326 div 78} frac{4}{17}
Therefore, the probability that two cards drawn from a well-shuffled deck are of the same suit is:
boxed{frac{4}{17}}
Conditional Probability Approach
Another way to approach this problem is through conditional probability. This method involves considering the first card selected and then finding the probability of the second card being of the same suit given that the first card has been selected without replacement.
Step 1: The First Card Can Be Any Card
The probability of the first card being any card is 1, as it can be any of the 52 cards in the deck.
52/52 1
Step 2: The Second Card Has to Be of the Same Suit as the First One
After the first card is drawn, there are 51 cards left in the deck. Of these, 12 cards belong to the same suit as the first card.
12/51 0.2353 or 23.53%
The combined probability of both events occurring is the product of the individual probabilities:
52/52 times 12/51 1 times 0.2353 0.2353 or 23.53%
This result is consistent with the previous calculation, confirming the simplified fraction 4/17.
Practical Implications and Variations
It is important to note that the problem can be modified based on the selection method. For instance, if the first card is revealed after the second card is selected, the probability changes. If the method involves looking at the first card and then ensuring the second card is of the same suit, the probability is 1. Conversely, if the method involves selecting a card with a different suit, the probability is 0. In all other cases, the probability will fall between these two extremes.
Conclusion
In summary, the probability of drawing two cards of the same suit from a well-shuffled deck of 52 cards is boxed{frac{4}{17}}. This result is derived using both combination and conditional probability methods, emphasizing the importance of understanding the problem's context in determining the appropriate approach.