Calculating the Probability of Drawing Three Red Cards Without Replacement from a Standard Deck
In this article, we will discuss the probability of drawing three red cards from a standard deck of cards without replacement. We will cover the step-by-step process, the concept of probability, and mathematical calculations involved in this scenario.
Introduction to Probability Basics
Probability theory is a crucial branch of mathematics that deals with the likelihood of events occurring. When dealing with a standard deck of 52 cards, each card is equally likely to be drawn during each draw, but since the cards are not replaced, the probability of drawing a specific card changes for each subsequent draw.
Standard Deck of Cards and Its Components
A standard deck of cards consists of 52 playing cards, including 26 red cards (13 diamonds and 13 hearts) and 26 black cards (13 spades and 13 clubs). The probability that the first card drawn is a red card is 26/52, or 1/2.
Step-by-Step Calculation
To calculate the probability of drawing three red cards without replacement, we need to consider the changing probabilities for each draw:
First Draw
The probability of drawing a red card first is:
26/52 1/2
Second Draw
After drawing one red card, there are now 25 red cards left and 51 cards total. The probability of drawing a second red card is:
25/51
Third Draw
After drawing two red cards, there are now 24 red cards left and 50 cards total. The probability of drawing a third red card is:
24/50 12/25
The overall probability of drawing three red cards without replacement is the product of these individual probabilities:
(26/52) * (25/51) * (24/50) 0.1176...
Alternative Method: Combinatorial Approach
Another way to calculate the probability is by using combinations. The number of ways to choose 3 red cards out of 26 is given by {26 choose 3}, and the number of ways to choose any 3 cards out of 52 is given by {52 choose 3}. Thus, the probability is:
{26 choose 3} / {52 choose 3} 2600 / 22100 0.1176...
Expected Value and Probability Distribution
For the expected value, we can use the hypergeometric distribution, which is particularly useful for sampling without replacement. Here, we have:
No of trials, n 3
Probability of success (drawing a red card), p 1/2
The probability of drawing a specific number of red cards can be calculated using the hypergeometric distribution formula:
Probability of drawing 0 red cards: {26 choose 0} * {26 choose 3} / {52 choose 3} 1 * 2600 / 22100 2/17 Probability of drawing 1 red card: {26 choose 1} * {26 choose 2} / {52 choose 3} 26 * 325 / 22100 13/34 Probability of drawing 2 red cards: {26 choose 2} * {26 choose 1} / {52 choose 3} 325 * 26 / 22100 13/34 Probability of drawing 3 red cards: {26 choose 3} * {26 choose 0} / {52 choose 3} 2600 * 1 / 22100 2/17These probabilities confirm the total probability distribution of drawing red cards.