Problem 7 - Calculus for Biology and Medicine

Question

You draw 3 cards from a standard deck of 52 cards without replacement. Let \(X\) denote the number of spades in your hand. Find the probability mass function describing the distribution of \(X\).

Step-by-Step Solution

Answer

The PMF for \(X\) is: \(P(X=0) = \frac{2926}{22100}, P(X=1) = \frac{7695}{22100}, P(X=2) = \frac{4389}{22100}, P(X=3) = \frac{286}{22100}\).

Step 1: Understand the Problem

We need to find the probability mass function (PMF) of the random variable \(X\), which represents the number of spades drawn from a standard deck of 52 cards in 3 draws without replacement.

Step 2: Identify Possible Values of X

The random variable \(X\) can take on the values 0, 1, 2, or 3 because the maximum number of spades we can draw in 3 cards is 3, and the minimum is 0.

Step 3: Calculate Probability for X=0

To draw 0 spades, all 3 cards must be non-spades. There are 39 non-spades in the deck. The probability is calculated as the ratio of the number of ways to draw 3 non-spades to the total number of ways to draw 3 cards:\[ P(X=0) = \frac{{\binom{39}{3}}}{{\binom{52}{3}}} \]

Step 4: Calculate Probability for X=1

To draw exactly 1 spade, we choose 1 spade from the 13 available and 2 non-spades from the 39 available. The probability is:\[ P(X=1) = \frac{{\binom{13}{1} \times \binom{39}{2}}}{{\binom{52}{3}}} \]

Step 5: Calculate Probability for X=2

To draw exactly 2 spades, we select 2 spades from the 13 available, and 1 non-spade from the 39 available. The probability is:\[ P(X=2) = \frac{{\binom{13}{2} \times \binom{39}{1}}}{{\binom{52}{3}}} \]

Step 6: Calculate Probability for X=3

To draw 3 spades, we have to choose all 3 cards from the 13 spades available. The probability is:\[ P(X=3) = \frac{{\binom{13}{3}}}{{\binom{52}{3}}} \]

Step 7: Compile the Probability Mass Function

We now have all probabilities needed to describe the PMF of \(X\):- \(P(X=0)\) for 0 spades,- \(P(X=1)\) for 1 spade,- \(P(X=2)\) for 2 spades,- \(P(X=3)\) for 3 spades.

Key Concepts

Random Variable

A "random variable" is a fundamental concept in probability theory. It's essentially a variable whose possible values are numerical outcomes of a random phenomenon. In the context of our problem with the deck of cards, the random variable

• Is denoted as \(X\), representing the number of spades drawn.
• Can take specific values, in this case, 0, 1, 2, or 3, corresponding to the number of spades drawn from the deck.

The power of using a random variable lies in its ability to translate everyday situations into mathematical form. This allows us to apply statistical techniques to understand and predict outcomes. You can think of a random variable as a bridge between our tangible activities and their abstract probabilities.

In this exercise, we examine the distribution of spades when drawing cards. Each possible outcome can be assigned a probability. This leads us directly to the Probability Mass Function (PMF), which provides a complete perspective of these probabilities, clearly showing us how likely each number of spades being drawn is.

Combinatorics

"Combinatorics" is a branch of mathematics that deals with counting combinations and permutations of sets. It plays a big role when working with problems involving the Probability Mass Function, especially in card games. Combinatorics helps us calculate the number of ways an event can occur.

In our card-drawing example,

• To find the number of ways to draw 3 cards from a deck, we use combinations. It's calculated using the binomial coefficient, denoted \(\binom{n}{k}\), which represents the number of ways to choose \(k\) elements from \(n\) elements without regard to order.
• To determine the possible arrangements of drawing cards, we calculate different combinations for spades and non-spades.

For instance, the formula \(\binom{39}{3}\) helps us compute how many ways to select 3 non-spades from the 39 available non-spade cards.

Understanding these combination calculations is crucial as they are the foundation upon which probability values for each scenario are computed.

Card Probability

"Card Probability" naturally involves determining the likelihood of drawing certain cards or combinations from a deck. This involves a thorough understanding of both the deck properties and the rules of probability – a perfect puzzle for combinatorics!

Consider a standard deck of 52 cards:

• There are 13 cards in each suit – spades, hearts, diamonds, and clubs.
• In probability problems, every card draw and the sequence of draws play into calculating the total outcomes.

When drawing without replacement, the compositional constant of the deck changes after each draw. This affects how we model the probabilities. For instance, drawing 1 spade has its probability given by the ratio of successful spade outcomes to total card outcomes possible, adapted for combinatorial nuances of spade and non-spade draws.

This approach clarifies the calculations behind probabilities such as \(P(X=1) = \frac{\binom{13}{1} \times \binom{39}{2}}{\binom{52}{3}}\), where the likelihood of drawing exactly one spade stems from the interplay of choosing from spades and non-spades in the deck.

Understanding these probabilities involves combining counting principles and probability theory, creating a cohesive understanding necessary for solving advanced probability problems.