Problem 26 - Algebra 2 and Trigonometry

Question

A group of friends are playing a game in which each player chooses a number from 1 to 6 and takes a turn tossing a die until the chosen number appears on the die. The required number of tosses is the person's score for that round of play. a. What is the probability that a player gets a score of 3 on the first round? b. What is the probability that a player gets a score of 5 on the first round? c. What is the probability of a score of \(n\) ? d. Show that the probabilities of scores that are consecutive integers from 1 to 5 form a geometric sequence.

Step-by-Step Solution

Answer

a. \(\frac{25}{216}\), b. \(\frac{625}{7776}\), c. \(\left(\frac{5}{6}\right)^{n-1} \times \frac{1}{6}\), d. Yes, they form a geometric sequence.

Step 1: Understanding the Problem

Each player's score is determined by how many tosses it takes for their chosen number to appear on a die, with possible outcomes ranging from 1 to 6. Our task is to find probabilities for various scores.

Step 2: Considering the Probability of Each Score

For each possible score, the die must not show the chosen number in the initial rolls and must show the chosen number on the final roll. The probability of rolling a specific number (e.g., the chosen one) on a fair six-sided die is \( \frac{1}{6} \). The probability of not rolling the chosen number is \( \frac{5}{6} \).

Step 3: Calculating Probability of Score 3

To score 3, the player needs to not get the chosen number twice and then get it on the third roll. The probability is \( \left(\frac{5}{6}\right)^2 \times \frac{1}{6} \).

Step 4: Calculating Probability of Score 5

To score 5, the player needs to not roll the chosen number four times, then roll it on the fifth turn. The probability is \( \left(\frac{5}{6}\right)^4 \times \frac{1}{6} \).

Step 5: Generalizing to Any Score n

To score \(n\), one must not roll the chosen number for \(n-1\) times, then roll it on the \(n\)th time. The probability of this happening is \( \left(\frac{5}{6}\right)^{n-1} \times \frac{1}{6} \).

Step 6: Showing a Geometric Sequence

The formula \( \left( \frac{5}{6} \right)^{n-1} \times \frac{1}{6} \) represents a geometric sequence as it is of the form \( ar^{n-1} \) where \( a = \frac{1}{6} \) and common ratio \( r = \frac{5}{6} \). Consequently, probabilities like \( n = 1 \) to \( n = 5 \) form a geometric progression.

Key Concepts

Geometric Sequence

In probability games like our dice game, we often encounter scenarios that form a geometric sequence. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant, called the common ratio.

In this dice game, the probabilities of scoring different numbers of turns (1, 2, 3, etc.) form a geometric sequence. For any score \( n \), we find that the probability is calculated as \( \left(\frac{5}{6}\right)^{n-1} \times \frac{1}{6} \). Here, \( \frac{5}{6} \) is the common ratio \( r \), and \( \frac{1}{6} \) acts as the first term \( a \) of the sequence.

• First term \( (a) = \frac{1}{6} \)
• Common ratio \( (r) = \frac{5}{6} \)

Recognizing such patterns can greatly simplify probability problems. It enables us to predict behaviors based on previously established mathematical principles, underscoring the importance of understanding sequences in mathematics.

Probability Calculation

Calculating probability involves determining the likelihood of an event occurring, which is often expressed as a fraction or percentage.

In our dice game, two types of probabilities are crucial:

• The probability of rolling a specific number, which is \( \frac{1}{6} \) for a fair 6-sided die.
• The probability of not rolling the chosen number, which in turn is \( \frac{5}{6} \).

To find the probability of needing exactly \( n \) rolls to get the chosen number, we first calculate the likelihood of not getting that number on the first \( n-1 \) rolls and then getting it on the \( n \)th roll. This is expressed as:
\[\left(\frac{5}{6}\right)^{n-1} \times \frac{1}{6}\]This mathematical expression allows us to systematically calculate the risk or chance of various outcomes. Understanding these calculations is critical for analyzing games of chance.

Dice Probability

Dice probability explores the calculated chance of rolling certain numbers with a die, which is foundational in hundreds of games worldwide like craps or Monopoly.

For a standard 6-sided die:

• Each face has a 1 out of 6 chance of appearing on a roll.
• The probability of rolling any specific number (say, number 4) is always \( \frac{1}{6} \).

In the context of our game, when a player rolls repeatedly to get their chosen number, it is essential to understand how each roll interacts. Initially, each roll has a probability of \( \frac{5}{6} \) of not showing the chosen number, but this accumulates until the chosen number appears.

Dice probability forms a core part of this setup, turning a simple game into a study of events and chance, highlighting the joy and complexity of probability.