Problem 18 - 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. \(P(E, \text { then } N),\) if replacement occurs

Step-by-Step Solution

Answer

The probability is \( \frac{1}{49} \).

Step 1: Identify Total Number of Tiles

The total number of tiles in the game is 7, as we have the tiles labeled with the letters E, T, F, U, N, X, and P.

Step 2: Determine the Probability of Choosing 'E' First

Since replacement occurs, the probability of picking the tile labeled 'E' first is \( \frac{1}{7} \).

Step 3: Determine the Probability of Choosing 'N' Second

Given that replacement occurs, after picking the first tile, the probability of again picking the tile labeled 'N' is also \( \frac{1}{7} \).

Step 4: Calculate The Combined Probability

The probability of both events occurring, choosing 'E' then 'N', is found by multiplying their individual probabilities due to independence: \( \frac{1}{7} \times \frac{1}{7} = \frac{1}{49} \).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arranging, and grouping items. In probability problems, it often helps us determine the number of ways something can occur. It is essential in calculating probabilities as it helps in understanding the different possible outcomes. For example, in choosing two tiles from a set of seven, combinatorics allows us to list and count all possible pairs that can be drawn.

Combinatorial techniques use formulas such as the factorial, combinations, and permutations. These tools allow you to figure out not just simple counts, but also more complicated structures like how many ways to arrange items or select items without regard to order.

In the context of the exercise, there are 7 tiles, and we are picking 2 of them. Since the order matters (picking E first and then N is different from N first then E), combinatorics informs us there are different methods, but because replacement is involved, we focus on independent selections instead.

Independent Events

In probability, events are independent if the outcome of one event does not affect the outcome of another.

Therefore, when two events are independent, the probability that both events occur is simply the product of their individual probabilities. In our exercise, choosing a tile labeled 'E' and then another labeled 'N' are independent events because the first tile is replaced before the second draw.

Thus, the events do not influence each other:

• The probability remains the same for each draw, as every tile is available each time.
• This independence allows for using multiplication to combine their probabilities effectively.

Understanding independence is crucial because it simplifies the calculation process, ensuring you only multiply probabilities when such conditions are met.

Replacement

Replacement refers to the action of putting a drawn item back before the next draw occurs. This action ensures that each draw is performed with the same set of possible outcomes. Replacement is critical in maintaining the independence of events in probability problems.

In the exercise, after selecting the first tile, it is put back with the others, keeping the total count at 7 for the second draw as well.

This method of replacement ensures that:

• The probability remains unaltered from one event to the next.
• Variables such as "which tiles are in the container" stay consistent.

Without replacement, each draw would alter the set of outcomes, impacting the probability of subsequent events and making the events dependent instead of independent.

Multiplying Probabilities

When dealing with independent events in probability, we often use the operation of multiplying probabilities. This operation helps us find the likelihood of multiple independent events occurring in sequence.

In our exercise, after confirming that choosing tiles is performed with replacement, each draw is independent of the other. Thus, to find the probability of drawing 'E' first and 'N' second, we multiply:

• The probability of drawing 'E', which is \( \frac{1}{7} \),
• By the probability of drawing 'N', also \( \frac{1}{7} \).

The combined probability is thus \( \frac{1}{49} \). This multiplication comes from the principle that, for independent events, the joint probability is the product of the probabilities of each event.