Problem 22 - Algebra 2

Question

The tiles \(E, T, F, U, N, X,\) and \(P\) of a word game are placed face down in the lid of the game. If two tiles are chosen at random, find each probability. \(P(2 \text { consonants), if no replacement occurs }\)

Step-by-Step Solution

Answer

The probability of drawing 2 consonants is \( \frac{10}{21} \).

Step 1: Identify the Consonants

In the given set of tiles \(\{E, T, F, U, N, X, P\}\), identify the consonants, which are \(\{T, F, N, X, P\}\). This provides us with 5 consonant tiles in total.

Step 2: Calculate the Total Outcomes

The total number of ways to choose 2 tiles from the 7 tiles \( (E, T, F, U, N, X, P) \) is given by the combination formula \( \binom{7}{2} \). Calculate \( \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \).

Step 3: Calculate the Favorable Outcomes

The number of favorable ways to choose 2 consonant tiles from the 5 consonant tiles \( (T, F, N, X, P) \) is given by \( \binom{5}{2} \). Calculate \( \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10 \).

Step 4: Find the Probability

Probability is calculated by dividing the number of favorable outcomes by the total outcomes. Therefore, \( P(2 \text{ consonants}) = \frac{10}{21} \).

Key Concepts

Combinations

Combinations are a way to determine how many different groups or subsets can be formed from a larger set without regard to the order of the elements. This is especially important in probability, where we often need to know how many possible outcomes there are.
In the context of our exercise, we need to know how to choose two tiles from a set of seven tiles: \ \[ \{E, T, F, U, N, X, P\} \]
We use a mathematical formula for combinations, denoted as \( \binom{n}{r} \), where \( n \) is the total number of items to choose from and \( r \) is the number of items to choose.
For our example, we calculate \( \binom{7}{2} \), because we are choosing 2 tiles from 7. The formula is:

• \( \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \)

This means there are 21 different ways to choose two tiles from the set.

Consonants

A consonant is a basic speech sound in which the breath is at least partly obstructed and which can include letters like T, F, N, X, and P. In contrast, vowels are letters like E and U.
For this exercise, we need to identify the consonants among the tiles we have:

• The consonant tiles are: \( \{T, F, N, X, P\} \)

There are 5 consonant tiles within our group.
These consonants are crucial because our goal is to calculate the probability of drawing two consonants when two tiles are randomly chosen. Understanding which tiles are consonants allows us to determine the favorable outcomes.

Probability without replacement

Probability without replacement refers to a situation where an item is not returned to the set after being selected. This means each draw changes the number of possible outcomes.
For calculating probability, we first determine the total number of possible selections, and then find the number of favorable outcomes.
In our problem, we first calculate the total outcomes, which is the combination of drawing any 2 tiles from 7: \ \[ \text{Total outcomes } = \binom{7}{2} = 21 \]
We then find the favorable outcomes, which are drawing 2 consonants from the 5 available: \ \[ \text{Favorable outcomes } = \binom{5}{2} = 10 \]
Finally, the probability is the ratio of the favorable outcomes to the total outcomes, which is:

• \( P(2 \text{ consonants}) = \frac{10}{21} \)

This probability is achieved without replacing any of the drawn tiles, meaning each choice impacts the next.