Problem 23 - 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. \(P(\text { selecting the same letter twice }),\) if no replacement occurs

Step-by-Step Solution

Answer

The probability is 0.

Step 1: Understand the Problem

We are tasked with finding the probability of selecting two identical tiles without replacement from the set of tiles {E, T, F, U, N, X, P}. Since each tile represents a different letter, none of them are identical. Thus, drawing the same letter twice without replacement is impossible.

Step 2: Identify the Total Number of Outcomes

First, calculate the total number of ways to choose any two tiles from the seven, which is a combination problem. The formula for combinations is given by \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items and \( r \) is the number of items to choose. Thus, the total number of combinations is \( \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \).

Step 3: Calculate the Probability of the Same Tile

Since no two tiles are identical, the number of successful outcomes, where you draw the same tile twice, is 0. Therefore, the probability of drawing the same tile is the number of successful outcomes divided by the total number of outcomes, i.e., \( \frac{0}{21} = 0 \).

Key Concepts

Combinations

When you have a set of items and want to know in how many different ways you can choose a certain number of them, you use combinations. Combinations are a fundamental part of probability and are often represented in mathematics by the binomial coefficient.

• They answer the question: "In how many ways can I select items?"
• The order of selection does not matter in combinations.

The formula for combinations is \[\binom{n}{r} = \frac{n!}{r!(n-r)!}\]where:

• \( n \) is the total number of items
• \( r \) is the number of items to choose
• \( n! \) represents the factorial of \( n \), which is the product of all positive integers up to \( n \)

In the original exercise, we used combinations to calculate the total number of ways to pick any two tiles from seven different tiles, which was found to be 21.

Sampling Without Replacement

Sampling without replacement means that once you select an item, you don't put it back before selecting another. This affects the probabilities because it changes the available pool of items to choose from with each draw.

• After drawing a tile, it is not returned to the set, so the total number decreases by one.
• Each subsequent selection is made from a smaller pool of items.

For the given problem, the tiles are drawn without replacement, implying that if you draw one tile, there will only be six tiles left for the second draw. This concept plays a crucial role in determining the total number of outcomes and affects the probability of selecting identical items.

Successful Outcomes

In probability, successful outcomes are the specific outcomes that fulfil the requirement of your probability question. They are the scenarios you're counting as "successes."

• A successful outcome for a probability problem is what you want to happen.
• In our example, it would mean drawing the same tile twice.

However, in this exercise, since the tiles are all unique and there is no replacement, it's impossible to draw the same tile twice. Therefore, the number of successful outcomes is zero. Understanding that sometimes the number of successful outcomes can be zero helps explain why some probabilities, like this one, can also be zero.

Total Outcomes

The total outcomes are the total number of possible different ways an event can occur. It's an essential component when calculating probabilities as it's the denominator in the probability fraction.

• Total outcomes include every possible pairing of tiles.
• For our example, 21 is the number of different combinations of 2 tiles chosen from the 7 available ones.

We found this by using the combination formula to compute all possible ways to draw two tiles from seven. Each pair is a different possible outcome, contributing to the denominator in the probability calculation. When determining the chance of any specific event, such as drawing two identical tiles, comparing successful to total outcomes is key.