Homework Help is Here – Start Your Trial Now!
Math
Probability
: A coin is tossed 4 times. Let the random variable X denote the # of tails that occur. a. List the outcomes of the experiment. (Note: n(S) = 2ª = 16) S= { b. Find the value assigned to each outcome by the random variable X. X = c. Find the event consisting of the outcomes to which a value of 2 has been assigned by X. E = {

: A coin is tossed 4 times. Let the random variable X denote the # of tails that occur. a. List the outcomes of the experiment. (Note: n(S) = 2ª = 16) S= { b. Find the value assigned to each outcome by the random variable X. X = c. Find the event consisting of the outcomes to which a value of 2 has been assigned by X. E = {

A First Course in Probability (10th Edition)
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...
See similar textbooks
icon
Related questions
Topic Video
Question
100%
**Coin Toss Experiment and Probability Distribution** When a coin is tossed 4 times, let the random variable \( X \) denote the number of tails that occur. **a. List the outcomes of the experiment.** The sample space \( S \) consists of all possible outcomes. Given \( n(S) = 2^4 = 16 \), \( S = \) the set of all sequences of heads (H) and tails (T) for 4 tosses. **b. Find the value assigned to each outcome by the random variable \( X \).** \( X = \) the number of tails in each sequence. **c. Find the event consisting of the outcomes to which a value of 2 has been assigned by \( X \).** \( E = \) the set of sequences where exactly 2 tails occur (e.g., HHTT, HTHT, etc.). **d. Create a probability distribution of \( X \).** \[ \begin{array}{c|c} x & P(X = x) \\ \hline 0 & \\ 1 & \\ 2 & \\ 3 & \\ 4 & \\ \end{array} \] **Note:** - \( 0 \leq P(x_i) \leq 1 \) - \( P(x_1) + P(x_2) + \dots + P(x_n) = 1 \) **e. Find:** a. \( P(X = 1) = \) b. \( P(X \geq 2) = \) c. \( P(1 \leq X \leq 3) = \) d. \( P(X < 4) = \) e. \( P(X = -4) = 0 \) (impossible event) **f. Draw a histogram of the probability distribution.** Create a histogram above with the x-axis labeled \( x \) and bars representing the probability \( P(X = x) \) for each value of \( x \). The sum of the heights of all bars should equal 1, representing the total probability.
Transcribed Image Text:**Coin Toss Experiment and Probability Distribution** When a coin is tossed 4 times, let the random variable \( X \) denote the number of tails that occur. **a. List the outcomes of the experiment.** The sample space \( S \) consists of all possible outcomes. Given \( n(S) = 2^4 = 16 \), \( S = \) the set of all sequences of heads (H) and tails (T) for 4 tosses. **b. Find the value assigned to each outcome by the random variable \( X \).** \( X = \) the number of tails in each sequence. **c. Find the event consisting of the outcomes to which a value of 2 has been assigned by \( X \).** \( E = \) the set of sequences where exactly 2 tails occur (e.g., HHTT, HTHT, etc.). **d. Create a probability distribution of \( X \).** \[ \begin{array}{c|c} x & P(X = x) \\ \hline 0 & \\ 1 & \\ 2 & \\ 3 & \\ 4 & \\ \end{array} \] **Note:** - \( 0 \leq P(x_i) \leq 1 \) - \( P(x_1) + P(x_2) + \dots + P(x_n) = 1 \) **e. Find:** a. \( P(X = 1) = \) b. \( P(X \geq 2) = \) c. \( P(1 \leq X \leq 3) = \) d. \( P(X < 4) = \) e. \( P(X = -4) = 0 \) (impossible event) **f. Draw a histogram of the probability distribution.** Create a histogram above with the x-axis labeled \( x \) and bars representing the probability \( P(X = x) \) for each value of \( x \). The sum of the heights of all bars should equal 1, representing the total probability.
# Educational Content on Probability Distributions ## Problem #5 Let \( X \) denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are rolled. Create a probability distribution of \( X \). ### Table for Probability Distribution of \( X \) | \( x \) | \( P(X = x) \) | |-------|--------------| | | | | | | | | | | | | | | | | | | ### Task In the space provided, draw a histogram of the probability distribution. ### Questions Find: a. \( P(X = 6) = \) b. \( P(X = 6 \text{ or } 12) = \) c. \( P(X < 6) = \) ### Types of Sets of \( X \): 1. Discrete, finite i.e. __________________ 2. Discrete infinite i.e. __________________ 3. Continuous i.e. __________________ --- ## Problem #28 Determine the probability distribution of the following: After the private screening of a new TV pilot, audience members were asked to rate the show on a scale of 1 - 10. From a group of 140 people, the following responses were obtained: ### Frequency and Probability Distribution Table | Rating \( x \) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |----------------------|---|---|---|---|----|----|----|----|----|----| | Frequency \( n(x) \) | 1 | 4 | 3 | 11| 23 | 21 | 28 | 29 | 16 | 4 | | \( P(X = x) \) | | | | | | | | | | | ### Explanation - **Histogram**: Create a histogram based on the probability values derived from the frequency distribution. - **Probability Computation**: Calculate \( P(X = x) \) by dividing each frequency \( n(x) \) by the total number of responses (140). Use this information to analyze and understand the distribution of the audience's ratings.
Transcribed Image Text:# Educational Content on Probability Distributions ## Problem #5 Let \( X \) denote the random variable that gives the sum of the faces that fall uppermost when two fair dice are rolled. Create a probability distribution of \( X \). ### Table for Probability Distribution of \( X \) | \( x \) | \( P(X = x) \) | |-------|--------------| | | | | | | | | | | | | | | | | | | ### Task In the space provided, draw a histogram of the probability distribution. ### Questions Find: a. \( P(X = 6) = \) b. \( P(X = 6 \text{ or } 12) = \) c. \( P(X < 6) = \) ### Types of Sets of \( X \): 1. Discrete, finite i.e. __________________ 2. Discrete infinite i.e. __________________ 3. Continuous i.e. __________________ --- ## Problem #28 Determine the probability distribution of the following: After the private screening of a new TV pilot, audience members were asked to rate the show on a scale of 1 - 10. From a group of 140 people, the following responses were obtained: ### Frequency and Probability Distribution Table | Rating \( x \) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |----------------------|---|---|---|---|----|----|----|----|----|----| | Frequency \( n(x) \) | 1 | 4 | 3 | 11| 23 | 21 | 28 | 29 | 16 | 4 | | \( P(X = x) \) | | | | | | | | | | | ### Explanation - **Histogram**: Create a histogram based on the probability values derived from the frequency distribution. - **Probability Computation**: Calculate \( P(X = x) \) by dividing each frequency \( n(x) \) by the total number of responses (140). Use this information to analyze and understand the distribution of the audience's ratings.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question

what's the answer for d?

Solution
Bartleby Expert
SEE SOLUTION
Knowledge Booster
Discrete Probability Distributions
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
Probability
ISBN:
9780134753119
Author:
Sheldon Ross
Publisher:
PEARSON
A First Course in Probability
A First Course in Probability
Probability
ISBN:
9780321794772
Author:
Sheldon Ross
Publisher:
PEARSON
SEE MORE TEXTBOOKS