Problem 28 - A First Course in Probability

Question

Suppose that an ordinary deck of 52 cards is shuffled and the cards are then turned over one at a time until the first ace appears. Given that the first ace is the 20th card to appear, what is the conditional probability that the card following it is the (a) ace of spades? (b) two of clubs?

Step-by-Step Solution

Answer

The conditional probability of the 21st card being the (a) ace of spades and (b) two of clubs, given that the first ace is the 20th card, is \(\frac{1}{51}\) for both cases.

Step 1: Find the total number of possible outcomes

Given that the first ace appears at the 20th card, there are 51 cards left in the deck (excluding the first ace). To find the total number of possible outcomes for the 21st card, we simply count the remaining cards, which is 51.

Step 2: Find the number of favorable outcomes

Since we want to find the probability of the 21st card being the ace of spades, there is only one favorable outcome – the card is the ace of spades.

Step 3: Calculate the conditional probability

Using the formula for conditional probability, we can calculate the probability of the 21st card being the ace of spades as: \(P(A|B) = \frac{P(A∩B)}{P(B)} = \frac{1}{51}\) Therefore, the conditional probability of the 21st card being the ace of spades is \(\frac{1}{51}\). Now, we will solve part (b) of the problem, which asks for the conditional probability of the 21st card being the two of clubs.

Step 1: Find the total number of possible outcomes (same as in part a)

The total number of possible outcomes is the same as in part (a) – there are 51 remaining cards in the deck after the first ace.

Step 2: Find the number of favorable outcomes

Since we want to find the probability of the 21st card being the two of clubs, there is only one favorable outcome – the card is the two of clubs.

Step 3: Calculate the conditional probability

Using the formula for conditional probability, we can calculate the probability of the 21st card being the two of clubs as: \(P(A|B) = \frac{P(A∩B)}{P(B)} = \frac{1}{51}\) Therefore, the conditional probability of the 21st card being the two of clubs is \(\frac{1}{51}\).

Key Concepts

Deck of Cards

The concept of a deck of cards is integral to understanding the problem at hand. A standard deck contains 52 cards, comprised of four suits: hearts, diamonds, clubs, and spades. Each suit has 13 ranks ranging from Ace to King.

A typical deck provides a tangible way to explore probability, as each card draw is an example of a random event. When shuffling the deck and drawing a card, each card has an equal chance of being selected, assuming a perfectly shuffled deck. This equal likelihood is central to many problems in probability theory, including the problem we are working on here.

Understanding the deck composition helps in identifying the total number of possible outcomes when drawing cards. Each draw changes the composition slightly, especially when cards are not replaced, just as in this exercise where one Ace was already drawn.

Probability Theory

Probability theory is the mathematical framework used to quantify the likelihood of various outcomes. In this context, we examine a scenario where we determine the probability of certain cards being drawn from the deck following a specific event.

A key concept in probability theory is the idea of independent and dependent events. In our card scenario, the draw of each card is a dependent event because the outcome changes with each card drawn. When an Ace appears in the 20th position, it directly affects the probabilities of subsequent draws.

The calculation of conditional probability becomes crucial here. It helps in understanding the likelihood of an event occurring given a certain condition has been met. It is calculated using the formula:

• \( P(A|B) = \frac{P(A∩B)}{P(B)} \)

This equation reflects the probability of event A occurring given that B is true. For the problem at hand, knowing the placement of Aces and the remaining cards assists in calculating the specific probabilities.

Calculating Outcomes

Calculating outcomes involves determining both the total and favorable scenarios. This is the foundation for finding probabilities. In our deck of cards problem, once the 20th card is an Ace, we have 51 cards left.

The total number of possible outcomes for the 21st card is simply the number of remaining cards in the deck, which is 51. This simplifies finding the likelihood of drawing a particular card next.

We then focus on favorable outcomes, which in this problem are either the ace of spades or the two of clubs following the first Ace. Since each of these cards exists once in the deck, there is just one favorable outcome for each scenario.

Simplifying the expression using the prior condition gives the probability of the card draw as \( \frac{1}{51} \) for both scenarios, because all remaining cards are equally likely to be the next card.