Problem 40 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Two cards are drawn without replacement from a wellshuffled deck of 52 cards. Let \(A\) be the event that the first card drawn is a heart, and let \(B\) be the event that the second card drawn is a red card. Show that the events \(A\) and \(B\) are dependent events.

Step-by-Step Solution

Answer

Events \(A\) and \(B\) are dependent events because the probability of both events happening together, \(P(A \cap B) = \frac{1}{8}\), is not equal to the product of the individual probabilities, \(P(A) \cdot P(B) = \frac{13}{102}\).

Step 1: Calculate probabilities of individual events

First, let's find the probability of each individual event. For event \(A\), the probability of drawing a heart as the first card is equal to the number of hearts divided by the total number of cards in the deck. There are 13 hearts in a deck of 52 cards, so: \(P(A) = \frac{13}{52} = \frac{1}{4}\) Next, let's find the probability of Event \(B\). A red card can either be a heart or a diamond, so there are 26 red cards in total. Since we have already drawn one card (referenced in Event A), there are now 51 cards left. \(P(B) = \frac{26}{51}\)

Step 2: Calculate the probability of both events happening together

Now, we need to find the probability of both events (\(A\) and \(B\)) happening together. If the first card drawn is a heart, there are 12 hearts left in the deck with 51 cards remaining. For the second card to be a red card (which includes hearts and diamonds), there are 25 red cards left. So, the probability of both events happening together is: \(P(A \cap B) = \frac{12}{51} \cdot \frac{25}{50}\) \(P(A \cap B) = \frac{1}{4} \cdot \frac{1}{2}\) \(P(A \cap B) = \frac{1}{8}\)

Step 3: Check if events \(A\) and \(B\) are dependent

Finally, let's check if events \(A\) and \(B\) are dependent by comparing the product of the individual probabilities with the probability of both events happening together. If these two values are not equal, then the events are dependent. \(P(A) \cdot P(B) = \frac{1}{4} \cdot \frac{26}{51} = \frac{13}{102}\) Since \(P(A \cap B) = \frac{1}{8}\) and \(P(A) \cdot P(B) = \frac{13}{102}\), we can conclude that: \(P(A \cap B) \neq P(A) \cdot P(B)\) This tells us that events \(A\) and \(B\) are dependent events.

Key Concepts

Probability Without Replacement

In probability theory, 'without replacement' means that once an outcome takes place, it cannot occur again. For example, when drawing cards from a deck, if a card is taken and not returned to the deck, the composition of the deck changes. This is essential in understanding dependent events since the probability of subsequent events can be influenced by the initial outcome.

Let's illustrate this with our exercise example. In the first draw, there are 13 hearts out of a total of 52 cards which gives us a probability of drawing a heart:
\( P(A) = \frac{13}{52} = \frac{1}{4} \). If we draw one heart and do not replace it, the deck now has 51 cards with only 12 hearts remaining for the second draw. Therefore, the probability of drawing another red card changes to: \( P(B|A) = \frac{26}{51} \) instead of \( P(B) = \frac{26}{52} \), because one card (a heart) is no longer there. This small change significantly impacts the outcome of compound events.

Conditional Probability

Conditional probability describes the likelihood of an event occurring provided that another event has already occurred. It is denoted as \( P(B|A) \), which reads as 'the probability of B given A'.

It's a cornerstone concept in determining whether events are dependent or independent. In the context of our exercise, after drawing the first heart (event A), the probability that the second card is red (event B) is based on the new deck condition. The formula for conditional probability in our case is:\( P(B|A) = \frac{26}{51} \), because the outcome of A influences the likelihood of B occurring.

Understanding conditional probability is vital for interpreting how one event alters the risk or odds of another within a series of occurrences.

Probabilities of Individual Events

The probability of an individual event refers to the chance that a specific outcome will occur in a single trial. This is irrespective of other events' outcomes. When events are independent, the outcome of one event does not affect or change the probability of another.

With regards to our example, each card draw from a full deck is an individual event. The initial chance of drawing a heart, event A, is \( P(A) = \frac{13}{52} = \frac{1}{4} \). However, the probability of these independent events starts to differ as we deal with sequential draws without replacement, leading to dependencies and the need to adjust calculations.

Probability of Compound Events

The probability of compound events is the likelihood of two or more events happening at the same time or in a sequence. It is determined by multiplying the probabilities of each individual event when they are independent, but when dealing with dependent events, we must take into account how the occurrence of one event impacts the others.

In the given example, we can observe this by finding the joint probability of drawing a heart and then a red card. The calculation yields \( P(A \cap B) = \frac{12}{51} \cdot \frac{25}{50} = \frac{1}{8} \), which differs from the product of the individual probabilities \( P(A) \) and \( P(B) \), signaling dependence between the events. This divergence shows that knowledge of previous outcomes is crucial for determining the combined probability of dependent events.