Problem 25 - Algebra 2

Question

Each of the numbers from 1 to 30 is written on a card and placed in a bag. If one card is drawn at random, what is the probability that the number is a multiple of 2 or a multiple of 3\(?\)

Step-by-Step Solution

Answer

The probability of drawing a card that is a multiple of 2 or 3 is \( \frac{2}{3} \).

Step 1: Understand the Total Outcomes

To solve a probability question, you need to know the total number of possible outcomes. In this case, every number from 1 to 30 is a possible outcome when drawing a card. So, the total number of possible outcomes is 30.

Step 2: Identify Multiples of 2

First, find all multiples of 2 between 1 and 30. These numbers are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, and 30. This gives us 15 numbers that are multiples of 2.

Step 3: Identify Multiples of 3

Next, find all multiples of 3 between 1 and 30. These numbers are 3, 6, 9, 12, 15, 18, 21, 24, 27, and 30. This results in 10 numbers that are multiples of 3.

Step 4: Find Overlapping Outcomes (Multiples of Both 2 and 3)

Identify numbers that are multiples of both 2 and 3 (i.e., multiples of 6). These are 6, 12, 18, 24, and 30, giving us 5 numbers.

Step 5: Apply the Inclusion-Exclusion Principle

To find the probability of drawing a multiple of 2 or 3, we use the formula: the number of multiples of 2 plus the number of multiples of 3 minus the number of multiples of both (6). Substituting the values gives: \[15 + 10 - 5 = 20\].

Step 6: Calculate the Probability

Now calculate the probability using the formula \( \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \). Substituting the known values gives: \[\frac{20}{30} = \frac{2}{3}\].

Key Concepts

Multiples

Multiples are the products you get from multiplying a given number by an integer. In our exercise, we're interested in the multiples of 2 and 3.
To find multiples of a number, simply keep adding the number to itself, or multiply it by whole numbers:

• For 2, the multiples are: 2, 4, 6, 8, and so on.
• For 3, the multiples are: 3, 6, 9, 12, and continue like that.

When you list out the multiples for each number, it becomes easier to see which numbers in a given range fit as multiples.
In our example, these multiples are the numbers we focus on between 1 and 30. Understanding multiples helps in narrowing down the favorable outcomes in probability problems.

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a key concept in probability and combinatorics. It allows us to calculate the probability of the union of two events.
In simpler terms, it helps in cases where two sets might overlap. Let's say we have sets A and B:

• Set A: Numbers that are multiples of 2
• Set B: Numbers that are multiples of 3

If we add the numbers from both sets directly, we end up counting the overlap twice. This is where the principle comes in:
The formula is given as \[|A \cup B| = |A| + |B| - |A \cap B|\]
By this formula, we add the two sets together and subtract the overlap. In our example exercise, this overlap are multiples of both 2 and 3, also known as multiples of 6.
By applying the inclusion-exclusion principle correctly, we achieve an accurate count of outcomes in our probability calculation.

Favorable Outcomes

In probability, favorable outcomes are the specific results we are interested in. These are the outcomes where the event occurs, in this case drawing a card with a number that is a multiple of 2 or 3.
It is vital to accurately determine these outcomes to correctly calculate probability.

• First, we found the multiples of 2 and 3 between 1 and 30.
• Then, we identified the overlap (multiples of 6).
• Finally, using inclusion-exclusion, we added the set of multiples of 2 and 3, subtracting the overlapping multiples of 6.

This precise count gives us the total favorable outcomes, which in this scenario is 20.
Once identified, we use these outcomes in our probability formula: \[\text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}\].
This gives us the likelihood of drawing a card that meets our requirement. Here, those outcomes yield a probability of \(\frac{2}{3}\).