A game is played using a four-sided wooden top, with one of four letters on each face, A, B, C, or D. Players compete for a pot of tokens. A player spins the top an- Cakes one of the following actions, depending on which letter faces up. If A faces up, the player neither adds nor subtracts tokens from the pot. - If B faces up, the player wins the entire pot of tokens. If C faces up, the player wins half of the tokens in the pot, rounding up if the number is odd. If D faces up, the player adds a token to the pot. Assume that each face of the top is equally likely to face upward and that the pot holds 26 tokens. Let the random variable X be the amount of tokens won by a layer. Thus, the range of X is {-1,0,13,26). Let the probability mass function be f(x) = P(X=x) and the cumulative probability function be F(x)=P(X ≤x). Find f(1) nd F(1). irst, find the probability mass function f(x). ype an ordered pair. Use a comma to separate answers as needed.) Ow find f(1). () = (Simplify your answer.)
A game is played using a four-sided wooden top, with one of four letters on each face, A, B, C, or D. Players compete for a pot of tokens. A player spins the top an- Cakes one of the following actions, depending on which letter faces up. If A faces up, the player neither adds nor subtracts tokens from the pot. - If B faces up, the player wins the entire pot of tokens. If C faces up, the player wins half of the tokens in the pot, rounding up if the number is odd. If D faces up, the player adds a token to the pot. Assume that each face of the top is equally likely to face upward and that the pot holds 26 tokens. Let the random variable X be the amount of tokens won by a layer. Thus, the range of X is {-1,0,13,26). Let the probability mass function be f(x) = P(X=x) and the cumulative probability function be F(x)=P(X ≤x). Find f(1) nd F(1). irst, find the probability mass function f(x). ype an ordered pair. Use a comma to separate answers as needed.) Ow find f(1). () = (Simplify your answer.)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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![**Educational Content on Cumulative Probability Function**
---
**Finding the Cumulative Probability Function**
To determine the cumulative probability function \(F(x) = P(X \leq x)\), choose the correct answer from the options provided.
**Options:**
**Option A:**
\[ F(x) = \begin{cases}
0 & x \leq -1 \\
1 & -1 \leq x \leq 26 \\
0 & 26 \leq x
\end{cases} \]
**Option B:**
\[ F(x) = \begin{cases}
0 & x \leq -1 \\
0.25 & -1 \leq x \leq 0 \\
0.5 & 0 \leq x \leq 13 \\
0.75 & 13 \leq x \leq 26 \\
1 & 26 \leq x
\end{cases} \]
**Option C:**
\[ F(x) = \begin{cases}
0.25 & -1 \leq x \leq 0 \\
0.5 & 0 \leq x \leq 13 \\
0.75 & 13 \leq x \leq 26
\end{cases} \]
**Option D:**
\[ F(x) = \begin{cases}
0 & x \leq -1 \\
0.25 & -1 \leq x \leq 0 \\
0.75 & 0 \leq x \leq 26 \\
1 & 26 \leq x
\end{cases} \]
**Now find \(F(1)\).**
\[ F(1) = \_\_\_\_\_ \] (Simplify your answer.)
---
**Detailed Explanation:**
To find the correct cumulative distribution function (CDF), we examine each option to see which appropriately represents the cumulative probabilities at given intervals.
- The correct CDF must start at 0 for \(x\) values outside the relevant range.
- It must increment correctly within the range where the step increase in probability should occur.
- It must sum to 1 for the highest \(x\) value in the range, indicating all probability is accounted for.
Analyze each option to ensure it adheres to the properties of CDFs:
- **Option B** seems to be](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7ffd19f8-d4bd-4354-b212-3cca5c840442%2Fb368cc4a-3fb4-4d29-8c81-c788ef575b8b%2Ftbhmo3p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content on Cumulative Probability Function**
---
**Finding the Cumulative Probability Function**
To determine the cumulative probability function \(F(x) = P(X \leq x)\), choose the correct answer from the options provided.
**Options:**
**Option A:**
\[ F(x) = \begin{cases}
0 & x \leq -1 \\
1 & -1 \leq x \leq 26 \\
0 & 26 \leq x
\end{cases} \]
**Option B:**
\[ F(x) = \begin{cases}
0 & x \leq -1 \\
0.25 & -1 \leq x \leq 0 \\
0.5 & 0 \leq x \leq 13 \\
0.75 & 13 \leq x \leq 26 \\
1 & 26 \leq x
\end{cases} \]
**Option C:**
\[ F(x) = \begin{cases}
0.25 & -1 \leq x \leq 0 \\
0.5 & 0 \leq x \leq 13 \\
0.75 & 13 \leq x \leq 26
\end{cases} \]
**Option D:**
\[ F(x) = \begin{cases}
0 & x \leq -1 \\
0.25 & -1 \leq x \leq 0 \\
0.75 & 0 \leq x \leq 26 \\
1 & 26 \leq x
\end{cases} \]
**Now find \(F(1)\).**
\[ F(1) = \_\_\_\_\_ \] (Simplify your answer.)
---
**Detailed Explanation:**
To find the correct cumulative distribution function (CDF), we examine each option to see which appropriately represents the cumulative probabilities at given intervals.
- The correct CDF must start at 0 for \(x\) values outside the relevant range.
- It must increment correctly within the range where the step increase in probability should occur.
- It must sum to 1 for the highest \(x\) value in the range, indicating all probability is accounted for.
Analyze each option to ensure it adheres to the properties of CDFs:
- **Option B** seems to be
Transcribed Image Text:**Title: Understanding Probability Mass and Cumulative Probability Functions through a Game**
**Game Description:**
A game is played using a four-sided wooden top, with one of four letters (A, B, C, or D) on each face. Players compete for a pot of tokens. A player spins the top and takes one of the following actions, depending on which letter faces up:
- **If A faces up**, the player neither adds nor subtracts tokens from the pot.
- **If B faces up**, the player wins the entire pot of tokens.
- **If C faces up**, the player wins half of the tokens in the pot, rounding up if the number is odd.
- **If D faces up**, the player adds a token to the pot.
Assume that each face of the top is equally likely to face upward and that the pot holds 26 tokens.
**Random Variable X:**
Let the random variable \( X \) be the amount of tokens won by a player. Thus, the range of \( X \) is \(\{-1, 0, 13, 26\}\).
**Probability Functions:**
Let the probability mass function be \( f(x) = P(X = x) \) and the cumulative probability function be \( F(x) = P(X \leq x) \).
**Task:**
Find \( f(1) \) and simplify your answer.
___
**Step-by-Step Calculation:**
**First, determine the probability mass function \( f(x) \):**
\( f(x) = \{ \} \)
(Type an ordered pair. Use a comma to separate answers as needed.)
**Next, find \( f(1) \):**
\( f(1) = \_\_\_\_ \) (Simplify your answer.)
In this scenario, understanding the probability of each outcome and properly defining the probability mass function is crucial. From here, one can proceed to calculate and interpret the cumulative probability function to gain a nuanced understanding of the game's dynamics.
**Diagrams/Graphs:**
The text does not mention any specific diagrams or graphs. If graphs illustrating the probability mass function or cumulative probability function were included, they would show:
- **Probability Mass Function (PMF) Graph:** A bar graph with \( X \) values on the x-axis (-1, 0, 13, 26) and their corresponding probabilities on the y-axis.
- **Cumulative Probability Function (
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