Problem 43 - Algebra 2

Question

REVIEW To begin a game, Nate must randomly draw a red, blue, green, or yellow game piece, and a tile from a group of 26 tiles labeled with all the letters of the alphabet. What is the probability that Nate will draw the green game piece and a tile with a letter from his name? $$ \begin{array}{ll}{\mathbf{F} \frac{1}{26}} & {\mathbf{H} \frac{3}{52}} \\\ {\mathbf{G} \frac{1}{13}} & {\mathbf{J} \frac{1}{2}}\end{array} $$

Step-by-Step Solution

Answer

The probability is \( \frac{1}{26} \).

Step 1: Identify the probability components

First, determine the probability of drawing a green game piece. There are 4 possible colors: red, blue, green, and yellow. Thus, the probability of drawing a green piece is \( \frac{1}{4} \).

Step 2: Identify letters in Nate's name

Nate's name consists of the letters N, A, T, and E. Therefore, the list of letters from his name is {N, A, T, E}, which means there are 4 unique letters from his name.

Step 3: Determine the probability of drawing a letter from Nate's name

Since there are 26 tiles and 4 tiles with letters from Nate's name, the probability of drawing a tile with a letter from his name is \( \frac{4}{26} \), which simplifies to \( \frac{2}{13} \).

Step 4: Calculate the compound probability

To find the probability of both events happening (drawing a green piece and a tile with a letter from Nate's name), multiply the individual probabilities from Steps 1 and 3: \( \frac{1}{4} \times \frac{2}{13} = \frac{2}{52} = \frac{1}{26} \).

Key Concepts

Compound Probability

Compound probability refers to finding the probability of two or more independent events happening together. In this context, the events are independent because the outcome of drawing a game piece does not affect the outcome of picking a tile from the alphabet set.

When calculating compound probability, you need to determine the probability of each individual event. Once you have those probabilities, you can combine them to find the overall probability.

• Identify each event that contributes to the combined outcome. Here, the events are drawing a green game piece and selecting a tile with a letter from Nate's name.
• Ensure that the events are independent. This means the occurrence of one does not influence the other, which is true in this case.

Through this process, you get closer to figuring out the chances of both events occurring together, using multiplication for a more concise solution.

Multiplying Probabilities

Multiplying probabilities is a key feature of finding the compound probability for independent events. Once you know the probability of each event happening separately, you multiply them to get the combined probability.

Here's how to multiply probabilities in this scenario:

• The probability of drawing a green game piece is given as \(\frac{1}{4}\).
• Once identified, the probability of getting a tile with a letter from Nate’s name is \(\frac{2}{13}\) (simplified from \(\frac{4}{26}\)).
• Multiply these probabilities using the formula for compound probability: \(\frac{1}{4} \times \frac{2}{13}\).

This multiplication gives \(\frac{2}{52}\), which represents the probability of both events occurring together. Such multiplication is crucial when dealing with more than one possibility happening at the same time.

Simplifying Fractions

Simplifying fractions is often the final step when calculating probabilities. It involves reducing the fraction to its simplest form so it's more straightforward and easier to understand.

This is usually accomplished by finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by it.

• For example, after multiplying the probabilities \(\frac{2}{52}\), look for a common divisor.
• The GCD of 2 and 52 is 2. Divide both by 2 to simplify \(\frac{2}{52}\) to \(\frac{1}{26}\).

The simplified fraction \(\frac{1}{26}\) then gives a straightforward probability, making it easier to understand the likelihood of drawing a green piece and a letter from Nate’s name. Always simplify fractions for clarity.