Question

Let us select five cards at random and without replacement from an ordinary deck of playing cards. (a) Find the pmf of \(X\), the number of hearts in the five cards. (b) Determine \(P(X \leq 1)\).

Short answer

The PMF of \(X\) can be calculated using combinatorics, and \(P(X\leq1)\) is simply the sum of the probabilities at \(X=0\) and \(X=1\) from the PMF.

Step-by-step solution

Count the Total Combinations

First, calculate the total number of different ways to draw 5 cards from a 52-card deck. This is given by \(C(52, 5)\), the binomial coefficient from combinatorics.

Compute the PMF

To compute the probability mass function of \(X\), the number of hearts in the five cards, for each possible value from 0 to 5, calculate the number of ways to draw that many hearts and the rest from non-hearts. This is given by the product of binomial coefficients, \(C(13, x) * C(39, 5-x)\) and dividing this number by the total combinations.

Calculate \(P(X\leq1)\)

To find the probability under 1 heart drawn, sum up the probabilities at \(X=0\) and \(X=1\). These are the values from the probability mass function computed in the previous step.

Key concepts

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc.

When we deal with problems like selecting cards from a deck, we're faced with a classic combinatorial problem: determining the number of different ways we can select a subset from a larger set. This problem types are often solved with the concepts of permutations and combinations, which are fundamental to understanding probability. In the case of the exercise provided, we're not just selecting cards; we're interested in selecting specific cards (hearts), which introduces the concept of combinations with specific conditions.

Understanding the basics of combinatorics is not just about counting but also about understanding how to structure the counts. This includes recognizing when certain items are indistinguishable, when order matters versus when it does not, and when replacements are allowed (or not), which alters the counting method used.

Binomial Coefficient

The binomial coefficient is a central element in combinatorics and appears frequently in probability theory. It is denoted as \(C(n, k)\) or sometimes as \(_nC_k\) or \({n \choose k}\), and it defines the number of ways to choose \(k\) elements out of a total of \(n\) distinct elements without regard to the order of selection. It is calculated using the formula:\[C(n, k) = \frac{n!}{k!(n-k)!}\]

For the provided exercise, the binomial coefficient helps us calculate the number of combinations of hearts and non-hearts that can be drawn, which is an essential step in constructing the probability mass function (PMF). The fact that cards are drawn without replacement makes the binomial coefficient the correct choice here, as it assumes that each element can be chosen only once, unlike permutations where elements can be repeated.

Understanding how the binomial coefficient works and how to calculate it is crucial when dealing with binomial distributions and probabilities where there are only two outcomes - like drawing a heart or not drawing a heart from a deck of cards.

Probability Theory

Probability theory provides the mathematical framework for quantifying uncertainty and is the foundation of statistical inference. Within this framework, the probability mass function (PMF) is a function that gives the probability that a discrete random variable is exactly equal to some value.

In the context of the exercise, the random variable \(X\) represents the number of hearts in the five drawn cards. The PMF describes the probability of each possible number of hearts. Calculating the PMF involves combining our knowledge of combinatorics (through the selection of combinations) and the binomial coefficients. These calculations give us a distribution of probabilities across all possible outcomes for \(X\).

Furthermore, probability theory not only allows us to find the likelihood of individual events but also to calculate cumulative probabilities, such as \(P(X \leq 1)\) which represents the probability of drawing either no hearts or one heart. This requires summing the PMFs for the relevant values of \(X\). Mastery of probability theory is thus essential for handling a wide range of quantitative problems in various fields from finance to engineering to social sciences.