Question

Let 13 cards be talsen, at random and without replacement, from an ordinary deck of playing cards. If \(X\) is the number of spades in these 13 cards, find the pmf of \(X\). If, in addition, \(Y\) is the number of hearts in these 13 cards, find the probability \(P(X=2, Y=5) .\) What is the joint pmf of \(X\) and \(Y ?\)

Short answer

The results have been calculated through hypergeometric distribution formulae. A table or graph displaying the pmf of \(X\) and joint pmf of \(X\) and \(Y\) would further elucidate the calculated values.

Step-by-step solution

Find the pmf of \(X\)

To find the pmf of \(X\), we'll be using the formula for the hypergeometric distribution: \[ P(X=k) = \frac{{C(13, k) \cdot C(39, 13-k)}}{{C(52, 13)}} \] Here, 13 is the total number of spades in the deck, 39 is the total number of other cards in the deck, 52 is the total number of cards in the deck, and \(C(n, k)\) denotes the number of combinations of \(n\) items taken \(k\) at a time. The pmf can be found for \(k = 0,1,...,13\).

Find \(P(X=2, Y=5)\)

To find this joint probability, we need to compute the probability that exactly 2 of the chosen cards are spades, and exactly 5 are hearts. \[ P(X=2, Y=5) = \frac{{C(13, 2) \cdot C(13, 5) \cdot C(26, 6)}}{{C(52, 13)}} \] \(C(13, 2)\) denotes the ways to pick 2 spades from 13, \(C(13, 5)\) denotes the ways to pick 5 hearts from 13, \(C(26, 6)\) denotes the ways to pick the remaining 6 cards from the 26 cards that are neither spades nor hearts.

Joint pmf of \(X\) and \(Y\)

The joint pmf of \(X\) and \(Y\) can be calculated as follows: \[ P(X=i, Y=j) = \frac{{C(13, i) \cdot C(13, j) \cdot C(26, 13-i-j)}}{{C(52, 13)}} \] This calculation should be performed for \(i = 0,1,...,13\) and \(j = 0,1,...,13-i\). This will yield the pmf for all possible values of \(X\) and \(Y\).

Key concepts

Hypergeometric Distribution

The hypergeometric distribution is a probability model often used to describe scenarios involving draws from a finite population without replacement. Imagine pulling a certain number of items from a container and being interested in how many belong to a specific group. That's precisely the type of question the hypergeometric distribution answers.

For example, if you have a standard deck of 52 playing cards and you draw 13 cards at random without putting any back ('without replacement'), the hypergeometric distribution can be used to determine the probability of drawing a specific number of spades. The key parameters for this distribution are the total number of items in the population (52 cards), the number of items of the type you're interested in (13 spades), and the number of items drawn (13 cards).

To compute the probability of drawing exactly 'k' spades using the hypergeometric distribution, we use combinatorial analysis to calculate the number of ways to select 'k' spades from the possible 13 and the number of ways to select the remaining cards from the non-spades. These combinatorial numbers are then divided by the total number of ways to draw any 13 cards from the deck.

Combinatorial Analysis

Combinatorial analysis involves the study of counting, arrangement, and combination of elements within a set in a systematic way. The fundamental tool in combinatorial analysis is the combinations formula, often denoted as 'C(n, k)' or sometimes by the 'choose' function in binomial coefficients. This function calculates the number of ways 'k' items can be selected from a set of 'n' distinct items without considering the order of selection.

Understanding Combinations

Imagine you have a plate with three different fruits: an apple, an orange, and a pear. If you want to pick two out of the three fruits, you might ask, 'How many different pairs can I have?' The answer lies in combinatorial analysis. Using the combinations formula, you can figure out that there are three possible pairs. This simple example scales up to more complex problems, like those found in the textbook exercise.

In the context of the exercise, combinatorial analysis is employed to determine the number of different ways to pick cards that are either spades or hearts, as well as the remaining cards from the deck.

Joint Probability

Joint probability is an extension of probability principles into multivariate cases, where we're interested in the probability of two or more events occurring simultaneously. It is represented as 'P(A and B)', indicating the likelihood of both events 'A' and 'B' happening at the same time.

In our cards example, the joint probability is used to answer questions like 'What's the probability of drawing exactly 2 spades and 5 hearts in the same 13-card hand?'. To calculate this, we need to consider all the individual probabilities together - the chances of picking exactly 2 spades, exactly 5 hearts, and the chances of picking the remaining cards from the rest of the deck. The individual probabilities are multiplied because, in a joint probability scenario, all events must occur in conjunction.

Understanding and calculating joint probability is crucial when outcomes are dependent on multiple conditions. It allows us to explore complex scenarios beyond single event probabilities, providing a fuller picture of the likelihood of various outcomes in multidimensional situations.