Problem 21 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A card is drawn from a well-shuffled deck of 52 playing cards. Let \(E\) denote the event that the card drawn is black and let \(F\) denote the event that the card drawn is a spade. Determine whether \(E\) and \(F\) are independent events. Give an intuitive explanation for your answer.

Step-by-Step Solution

Answer

Events \(E\) and \(F\) are independent because the probability of their intersection equals the product of their individual probabilities, that is, \(P(E \cap F) = P(E) \cdot P(F) = \frac{13}{52}\). Intuitively, the probability of drawing a spade is not affected by the fact that the card is black, as there's still a \(\frac{13}{52}\) chance of drawing a spade regardless of the card's color.

Step 1: Identify the probabilities of events E and F

There are 26 black cards in a standard deck of 52 playing cards (13 spades and 13 clubs), and there are 13 spades in total. To find the probability of each event, we'll divide the number of favorable outcomes by the total number of outcomes in the sample space. \(P(E) = \) Probability that the card drawn is black = \(\frac{26}{52}\) \(P(F) = \) Probability that the card drawn is a spade = \(\frac{13}{52}\)

Step 2: Identify the probability of the intersection of events E and F

The intersection of events \(E\) and \(F\), denoted as \(E \cap F\), is the event that the card drawn is both black and a spade. Since all spades are black, this event is equivalent to the event that the card drawn is a spade. \(P(E \cap F) =\) Probability that the card drawn is black and a spade = \(\frac{13}{52}\)

Step 3: Determine if E and F are independent

Events \(E\) and \(F\) are independent if the probability of their intersection equals the product of their individual probabilities: \(P(E) \cdot P(F) = \frac{26}{52} \cdot \frac{13}{52} = \frac{13}{52}\) Comparing, we have: \(P(E \cap F) = P(E) \cdot P(F)\)

Step 4: Conclusion

Since the probabilities satisfy the condition of independence - the probability of the intersection of events E and F equals the product of their individual probabilities - events \(E\) and \(F\) are independent events. Intuitively, this makes sense because the probability of drawing a spade is not affected by the fact that the card is black; we would still have a 13 in 52 chance of drawing a spade regardless of the card's color.

Key Concepts

Probability Theory

Understanding the fundamentals of probability theory is essential to grasping how we can predict the likelihood of certain events. Probability is the measure of the likelihood that an event will occur, quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. A key concept here is the sample space, which encompasses all possible outcomes of a random experiment. For instance, the sample space for a die roll includes the outcomes {1, 2, 3, 4, 5, 6}.

When we calculate the probability of an event, we use a simple formula: \, \(\text{Probability of an event} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). Using the card-drawing exercise as an example, there are 52 total outcomes (one for each card), making the probability of drawing any single card 1/52.

Conditional Probability

Conditional probability is concerned with the likelihood of an event occurring given that another event has already taken place. Referenced with the notation \(P(A | B)\), it translates to 'the probability of event \(A\) occurring given that event \(B\) has occurred'. Clarifying this aspect of probability can untangle the interrelation of events. For example, if we wanted to find the probability of drawing an ace from a deck given that the drawn card is black, it would require us to consider only the black cards when calculating our odds, not the entire deck.

In the solved problem, we would've used conditional probability if we were asked to find the chance of drawing a spade given the card is black. Since all spades are already black, the probability remains unchanged; therefore, our initial calculation of individual probabilities suffices.

Sample Space in Probability

The concept of sample space is integral to probability calculations. It represents the set of all possible outcomes in a random experiment and is often denoted by the symbol \(S\). In our card example, the sample space contains 52 elements, each representing a distinct card from the deck. Understanding the makeup of the sample space enables us to define events, which are subsets of the sample space, and calculate their probabilities accurately. The size and nature of the sample space determine the complexity of the probability calculations, which can range from very simple to extremely intricate, as seen in more complex games involving multiple events and players.

Combinatorics

Combinatorics is the branch of mathematics dealing with counting, combination, and permutation of sets. Important in probability, it helps us calculate the number of ways events can occur, thus significantly contributing to the determination of their probability. There are two fundamental principals in combinatorics: the addition rule, used when events cannot occur simultaneously, and the multiplication rule, used when events are independent. In our card problem, understanding that we have 13 spades, each of which is a black card, reflects basic combinatorial principles at work. More generally, combinatorics allows us to count without enumerating every single possibility, thus simplifying complex probability problems.