Problem 21 - Calculus for Biology and Medicine

Question

21\. You pick 2 cards from a standard deck of 52 cards. Find the probability that the second 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 second card is an ace is slightly higher than the probability that the first card is an ace.

Step 1: Understand the Scenario

Consider a standard deck of 52 cards. You will pick one card, then a second card without replacing the first. We need to find the probability that the second card drawn is an ace.

Step 2: Calculate Probability of Second Card Being an Ace

There are 4 aces in a deck of 52 cards. When the second card is drawn, 51 cards remain. Thus, the probability that the second card is an ace is \( \frac{4}{51} \).

Step 3: Calculate Probability of First Card Being an Ace

Initially, there are also 4 aces in a deck of 52 cards. So, the probability that the first card is an ace is \( \frac{4}{52} = \frac{1}{13} \).

Step 4: Compare Both Probabilities

Comparing the two probabilities, \( \frac{4}{51} \) (second card) and \( \frac{1}{13} \) (first card). Convert \( \frac{1}{13} \) to have the same denominator: \( \frac{1}{13} = \frac{4}{52} \). This makes the first probability slightly smaller since \( \frac{4}{51} > \frac{4}{52} \).

Key Concepts

Standard Deck of Cards

A standard deck of cards is an essential tool in understanding probability in card games. This deck consists of 52 cards, divided into four suits: Hearts, Diamonds, Clubs, and Spades. Each suit contains 13 cards, which are numbered from 2 through 10, followed by the Jack, Queen, King, and Ace. When we explore probability scenarios in card games, we must take into account the composition of this deck.

Knowing these details is crucial when calculating the probability of drawing certain types of cards. For example, each suit contains one ace, so there are four aces in total. This uniformity is key to solving probability problems in card games as it provides a predictable and consistent framework from which probabilities can be calculated.

Probability Comparison

In probability comparison, we investigate how likely different events are relative to one another. In the context of card games, we can analyze the probabilities of drawing certain cards in sequence.

For instance, when comparing the probability of drawing an ace on the first draw versus the second, we observe some interesting differences. Initially, there are 4 aces out of 52 cards, giving a probability of \( \frac{1}{13} \) for the first card being an ace. However, after one card is drawn, only 51 remain. If the first card wasn't an ace, the probability of the second card being an ace then becomes \( \frac{4}{51} \).

Even though both probabilities involve drawing an ace, different denominators result from different scenarios. Thus, through comparison, we illustrate how probabilities can shift based on prior outcomes and remaining possibilities.

Combinatorics in Probability

Combinatorics is a field of mathematics focused on counting, arranging, and combination possibilities. It plays a crucial role in determining probabilities in card games, especially when multiple draws are involved.

When using combinatorics, it's important to understand the idea of choosing without replacement, which affects the total number of outcomes. For example, if you're calculating the probability that the second card drawn is an ace, you have to consider that only 51 cards are left, thus there are fewer options to choose from.

This concept underpins many card game probability problems, as we must adjust our calculations to reflect the changing landscape of possible outcomes. Combinatorics helps us break down complex probability tasks into manageable parts, allowing for clear, accurate predictions in games.