Problem 41 - A First Course in Probability

Question

A deck of cards is shuffled and then divided into two halves of 26 cards each. A card is drawn from one of the halves; it turns out to be an ace. The ace is then placed in the second half-deck. The half is then shuffled, and a card is drawn from it. Compute the probability that this drawn card is an ace. Hint: Condition on whether or not the interchanged card is selected.

Step-by-Step Solution

Answer

The probability of drawing an ace from the second half-deck after an ace has been added to it is \(\frac{4}{27}\).

Step 1: Identify the possible outcomes

There are two possible outcomes when a card is drawn from the second half-deck: either the card is an ace, or it is not an ace. We need to find the probability of drawing an ace.

Step 2: Calculate the probability of finding an ace in the second half-deck before moving an ace to it

Before the ace is added to the second half-deck from the first half-deck, there are \(3\) aces in the second half-deck and a total of \(26\) cards. Thus, the probability of drawing an ace from the second half-deck before moving the ace is given by: \[P(Ace \: in \: 2nd \: Half - Deck) = \frac{3}{26}\]

Step 3: Calculate the probability of drawing the interchanged ace from the second half-deck

After the ace is added to the second half-deck, there are now \(4\) aces and a total of \(27\) cards. The added ace has a probability of \(\frac{1}{27}\) of being drawn.

Step 4: Calculate the probability of not drawing the interchanged ace

Since the probability of drawing the interchanged ace is \(\frac{1}{27}\), the probability of not drawing the interchanged ace is: \[P(Not \: drawing \: interchanged \: Ace) = 1 - \frac{1}{27} = \frac{26}{27}\]

Step 5: Use conditional probability to find the probability of drawing an ace

We find the probability of drawing an ace from the second half-deck after an ace has been added by conditioning on whether or not the interchanged ace is selected. The probability of drawing an ace is the sum of the probabilities of drawing the interchanged ace and drawing one of the original three aces, given that the interchanged ace is not selected: \[P(Drawing \: an \: Ace) = P(Interchanged \: Ace) + P(Ace \: in \: 2nd \: Half - Deck | Not \: drawing \: interchanged \: Ace) \cdot P(Not \: drawing \: interchanged \: Ace)\] Substituting the probabilities we calculated earlier: \[P(Drawing \: an \: Ace) = \frac{1}{27} + \frac{3}{26} \cdot \frac{26}{27} = \frac{1}{27} + \frac{3}{27} = \frac{4}{27}\] Therefore, the probability of drawing an ace from the second half-deck after an ace has been added to it is \(\frac{4}{27}\).

Key Concepts

Understanding Probability Theory

Probability theory allows us to calculate the likelihood of various outcomes when dealing with random events. In this context, it is particularly useful to understand how likely it is for a card drawn from a shuffled deck to be an ace. Probability measures the uncertainty of outcomes. For example, if you need to find the probability that you will draw an ace from a deck of cards, it involves understanding both the favorable outcomes and the total number of outcomes.

To begin, if a deck has 52 cards and there are 4 aces, the probability of drawing an ace initially is the ratio of aces to the total cards, or \(\frac{4}{52} = \frac{1}{13}\). When a condition affects the event, such as moving cards between decks, conditional probability is employed to adjust these calculations.

Conditional probability considers how the likelihood of an event changes given the occurrence of another related event. This is key in our exercise where an ace is moved between halves and the probability is conditioned on whether or not the moved ace is selected. Using conditional probability allows for precise calculations in real-world applications, making it an essential concept in probability theory.

The Art of Card Shuffling

Card shuffling is a random process meant to rearrange the cards in a deck. It ensures that each card has an equal probability of being drawn. This is crucial in games of chance where fairness is a priority.

In this exercise, shuffling is the first step. The full deck is shuffled to start, dividing the cards into two groups of 26. After an ace is drawn from this grouping, it is placed into the second group and shuffled again.

Good shuffling minimizes order and ensures randomness. Many techniques exist:

• Riffle shuffle: splitting the deck into two and interleaving them.
• Overhand shuffle: cards are repeatedly taken from the top and placed into the hand.
• Wash shuffle: cards are gathered in a pile and mixed by spreading.

These methods help achieve a well-randomized deck, crucial for calculating unbiased probabilities in events such as drawing cards. Understanding card shuffling enhances comprehension of randomness in probability theory and its applications in card games.

Calculating Expected Value

Expected value offers a way to quantify the average outcome of a random event over time. In probability, it helps anticipate the average result if an action were repeated numerous times. The expected value is calculated using probabilities of outcomes and their corresponding values.

Suppose you repeatedly draw a card from a shuffled deck. Calculating the expected value of drawing an ace involves multiplying the probability of the ace by its "value" or weight in this scenario. Using probabilities from the exercise, if drawing an ace had a probability of \(\frac{4}{27}\), the expected value can be considered as roughly reflecting the percentage, or chance, of drawing an ace over multiple draws.

This concept is beneficial in many fields, such as finance, when assessing the potential returns of investments, or in gaming, predicting outcomes to determine strategic benefits. In the exercise, achieving an expected value illustrates the chances of drawing an ace given conditions applied, thus providing a better understanding of probability in practice.