Problem 46 - Algebra 2

Question

Two cards are drawn from a standard deck of cards. Find each probability. P(heart, then club) if no replacement occurs

Step-by-Step Solution

Answer

The probability is \( \frac{1}{22} \).

Step 1: Understand the problem

We are asked to find the probability of drawing a heart first and then a club from a standard deck, without replacement. A standard deck has 52 cards: 13 hearts and 13 clubs.

Step 2: Identify outcomes for the first event

The probability of drawing a heart from the deck is calculated by finding the ratio of hearts to the total cards. There are 13 hearts in a deck of 52 cards. Thus, the probability of drawing a heart is \( \frac{13}{52} \).

Step 3: Determine outcomes for the second event

Once a heart is drawn, there are 51 cards left with 13 clubs remaining. Therefore, the probability of drawing a club after the heart is \( \frac{13}{51} \).

Step 4: Calculate the combined probability

The probability of both events happening in sequence (drawing a heart and then a club) is the product of their individual probabilities. Hence, the probability is \( \frac{13}{52} \times \frac{13}{51} = \frac{169}{2652} \).

Step 5: Simplify the probability

To get the simplified form, divide both the numerator and the denominator by their greatest common divisor (169), resulting in \( \frac{1}{22} \).

Key Concepts

Deck of Cards

A standard deck of cards is a key player in probability exercises. It has a total of 52 cards. These are split evenly into four suits: hearts, diamonds, clubs, and spades. Each suit contains 13 cards. This setup is essential for many probability calculations because each card is unique.When dealing with deck-based problems, remember:

• 13 hearts
• 13 diamonds
• 13 clubs
• 13 spades

Understanding this structure helps in calculating the probability of drawing specific types of cards. For instance, knowing that 13 of the 52 cards are hearts makes it straightforward to understand why the probability of drawing a heart is \( \frac{13}{52} \). This fundamental knowledge is crucial for solving and understanding card-related probability problems.

Events Sequence

In probability, the sequence of events is crucial. This refers to the order in which events happen and how each event affects subsequent events. Take the scenario of drawing cards from a deck. If you're drawing two cards in sequence, what you get the first time affects what you can possibly get the next time. For our current problem:

• The first event is drawing a heart.
• The second event follows, which is drawing a club.

Each step has its own probability, and to find the probability of this particular sequence of events, we multiply the probabilities of the individual events together. This is known as the rule of multiplication for independent events, although in situations like this where there is no replacement, events are not independent.

Without Replacement

"Without replacement" means that once an item is used, it can't be used again. In card exercises, if you draw a card and don't put it back, there are fewer cards left for the next draw. This changes the probability of the next event. For example, if you draw a heart from a 52-card deck without returning it:

• You are left with 51 cards.
• The number of hearts decreases to 12 (if that was what you drew), affecting future draws.

This is why in the second draw, you calculate the probability of drawing a club out of 51 and not 52 cards. Problems "without replacement" often require adjusting the denominator and sometimes the numerator on each step to accurately calculate probabilities.

Simplifying Fractions

Simplifying fractions is an important skill in probability problems to ensure clarity and accuracy. It involves reducing a fraction to its simplest form, where the numerator and the denominator have no common divisor other than 1.In this exercise, once you have the probability result \( \frac{169}{2652} \), you simplify it by finding the greatest common divisor (GCD) of 169 and 2652. Here, the GCD is 169, allowing:\[\frac{169}{2652} = \frac{169 \div 169}{2652 \div 169} = \frac{1}{22}\]Simplifying makes the fraction more understandable and often easier to compare with other probabilities. It also provides the final probability in the neatest form possible, making it clear and precise for any analysis or further use.