Problem 22 - Calculus for Biology and Medicine

Question

You pick 3 cards from a standard deck of 52 cards. Find the probability that the third card is an ace. Compare this with the probability that the first card is an ace.

Step-by-Step Solution

Answer

The probability that the first card is an ace is \( \frac{1}{13} \), and the probability that the third card is an ace is also \( \frac{1}{13} \). Both probabilities are the same.

Step 1: Determine Total Possibilities

A standard deck has 52 cards. For any of the 3 draws, there are still 52 cards in the deck. So, when you draw one card from the deck, there are 52 possible outcomes.

Step 2: Probability of an Ace on the Third Draw

By the time you draw the third card, no matter what happened on the first two draws, there are still 52 ways to draw a card from the deck and 4 of these are aces, just as at the beginning of the deck. This means the probability that the third card is an ace remains unchanged at \( \frac{4}{52} \) or \( \frac{1}{13} \).

Step 3: Probability of an Ace on the First Draw

When drawing the first card, there are still 52 total cards in the deck and 4 of them are aces. So, the probability that the first card is an ace is also \( \frac{4}{52} \) or \( \frac{1}{13} \).

Step 4: Compare the Probabilities

The probability that the first card is an ace and the probability that the third card is an ace are both \( \frac{1}{13} \). The order of drawing does not affect the probability, as each card draw from a complete deck is an independent event.

Key Concepts

Independent Events

In card games, understanding the concept of independent events is crucial. An independent event means that the occurrence of one event does not affect the outcome of another event. Simply put, what happens in one scenario doesn't change what happens in another.

When drawing cards from a complete, unaltered deck, each draw is an independent event. For example, if you draw one card, replace it, and then draw again, the second draw has the same probability distribution as the first. Even if you do not replace the card, analyzing the scenario mathematically considers the cards drawn independently until you change the total number of possibilities. In our problem, since the deck remains complete during both the first and subsequent draws, it magnifies the independence of each event. No matter if you pick the first, second, or third card, the probability remains consistent.

• This is because none of the draws have affected the remaining cards in the deck until the completion of the draw.
• Since probabilities are abstractly considered based on a complete deck, the dependency is not factored in until a card is definitively not replaced.

Deck of Cards

To decode probability in card games, understanding a deck of cards is fundamental. A standard deck comprises 52 unique cards divided into 4 suits: hearts, diamonds, clubs, and spades.

Each suit contains 13 cards, which include numbers from 2 to 10, and face cards: jack, queen, king, and ace. The uniformity of suits means each ace is equally likely to be drawn. Calculating probabilities typically involves understanding how many desired outcomes exist within these parameters.

• With every draw, you can choose from any of these 52 cards.
• Ace cards form the special interest here, with a total of 4 aces spread equally in the suits.
• To balance outcomes, each card maintains an equal baseline chance of being selected if not replaced.

Recognizing the role of this division helps clarify why the probability calculations remain consistent over different positions during a card game.

Probability Calculation

Probability calculation helps us figure out the chance of an event happening. This calculation is especially useful in card games.

Probability is defined as the number of successful outcomes divided by the total number of possible outcomes. For any card draw, the event is finding an ace from the deck. You find probability using a simple formula:

• Probability of an event = \( \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}} \)

In this exercise, for both the first and third card draws, the favorable outcome is drawing an ace and possible outcomes remain at 52 (since the deck's structure stays unchanged), maintaining the same probability.

This is expressed as:

• Probability of an ace being drawn first or third = \( \frac{4}{52} = \frac{1}{13} \)

The concept of probability remains stable when viewed in such isolated conditions, providing clarity and predictability regardless of draw sequence.