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Chapter 4: Problem 11

Use the following information to answer the next four exercises: Ellen has music practice three days a week. She practices for all of the three days 85% of the time, two days 8% of the time, one day 4% of the time, and no days 3% of the time. One week is selected at random. Construct a probability distribution table for the data.

Short Answer

Expert verified
The probability distribution table is: X=0:0.03, X=1:0.04, X=2:0.08, X=3:0.85.

Step by step solution

01

Identify Possible Values of the Random Variable

In this problem, the random variable X represents the number of days that Ellen practices music in a week. The possible values are 0, 1, 2, or 3 based on the given conditions.
02

Extract Probabilities for Each Value

From the given information, the probabilities for each value of X are provided: - Probability of practicing all 3 days: P(X=3) = 85% = 0.85 - Probability of practicing 2 days: P(X=2) = 8% = 0.08 - Probability of practicing 1 day: P(X=1) = 4% = 0.04 - Probability of practicing 0 days: P(X=0) = 3% = 0.03.
03

Construct the Probability Distribution Table

Using the values and probabilities identified, construct the table. The table should include the possible values of the random variable X and their corresponding probabilities:\[\begin{array}{|c|c|}\hlineX & P(X) \\hline0 & 0.03 \1 & 0.04 \2 & 0.08 \3 & 0.85 \\hline\end{array}\]
04

Verify that Probabilities Sum to 1

Ensure that the sum of all probabilities in the distribution is 1 to confirm it is a valid probability distribution:\[P(X=0) + P(X=1) + P(X=2) + P(X=3) = 0.03 + 0.04 + 0.08 + 0.85 = 1.00\]This sum confirms that the probabilities are correctly distributed.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Random Variable
In the world of probability, a random variable is a key concept that helps us measure outcomes. Think of it as a mathematical function that assigns a numerical value to each possible outcome of a random event. In the case of Ellen's music practice schedule, the random variable \( X \) represents the number of days she practices in a week. For this situation:
  • 0 days: Ellen does not practice at all.
  • 1 day: Ellen practices one day in a week.
  • 2 days: Ellen practices on two separate days.
  • 3 days: Ellen manages to practice every single available day.
These values capture all possible outcomes of Ellen's weekly practice schedule, making \( X \) a discrete random variable since it can only take on specific, isolated values.
Probability Table
A probability table is a structured way to present the likelihood of various outcomes of a random variable. In Ellen's scenario, the probability table helps us see at a glance how often she practices music across different days in a week. The table we create should reflect:
  • The possible outcomes (0, 1, 2, or 3 days).
  • The probability associated with each outcome, which shows how likely each is to occur.
For Ellen:
  • Practicing all three days has a probability of 0.85.
  • Practicing two days has a probability of 0.08.
  • Practicing one day has a probability of 0.04.
  • Practicing none of the days happens 0.03 of the time.
Constructing a probability table effectively organizes this data, helping anyone understand how frequently each outcome occurs.
Valid Probability Distribution
A probability distribution is a complete description of all the probabilities of a random variable鈥檚 possible outcomes. For it to be valid, two main conditions must be satisfied:
  • All probabilities must be non-negative and range between 0 and 1.
  • The sum of all probabilities must equal 1.
In Ellen's practice schedule, the probabilities are already given:
  • 0.85 for 3 days.
  • 0.08 for 2 days.
  • 0.04 for 1 day.
  • 0.03 for 0 days.
To check if these form a valid probability distribution, you add them together:\[ 0.85 + 0.08 + 0.04 + 0.03 = 1.00 \]Since they add up to 1, this confirms that the probability distribution is valid. Understanding these conditions can ensure you correctly interpret and construct probability distributions in any scenario.

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Most popular questions from this chapter

Use the following information to answer the next four exercises. Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about 4%. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu. Find the probability that at least four of the 25 patients actually have the flu.

Find the standard deviation. $$\begin{array}{|c|c|c|c|}\hline x & {P(x)} & {x^{*} P(x)} & {(x-\mu)^{2} P(x)} \\ \hline 2 & {0.1} & {2(0.1)=0.2} & {(2-5.4)^{2}(0.1)=1.156} \\\ \hline 4 & {0.3} & {4(0.3)=1.2} & {(4-5.4)^{2}(0.3)=0.588} \\ \hline 6 & {0.4} & {6(0.4)=2.4} & {(6-5.4)^{2}(0.4)=0.144} \\ \hline 8 & {0.2} & {8(0.2)=1.6} & {(8-5.4)^{2}(0.2)=1.352} \\ \hline \end{array}$$

Use the following information to answer the next two exercises: The probability that the San Jose Sharks will win any given game is 0.3694 based on a 13-year win history of 382 wins out of 1,034 games played (as of a certain date). An upcoming monthly schedule contains 12 games. It has been estimated that only about 30% of California residents have adequate earthquake supplies. Suppose you randomly survey 11 California residents. We are interested in the number who have adequate earthquake supplies. a. In words, define the random variable \(X.\) b. List the values that \(X\) may take on. c. Give the distribution of \(X . X \sim\) ___ (___,____) d. What is the probability that at least eight have adequate earthquake supplies? e. Is it more likely that none or that all of the residents surveyed will have adequate earthquake supplies? Why? f. How many residents do you expect will have adequate earthquake supplies?

Identify the mistake in the probability distribution table. $$\begin{array}{|c|c|c|}\hline x & {P(x)} & {x^{\star} P(x)} \\ \hline 1 & {0.15} & {0.15} \\ \hline 2 & {0.25} & {0.50} \\ \hline 3 & {0.30} & {0.90} \\\ \hline 4 & {0.20} & {0.80} \\ \hline 5 & {0.15} & {0.75} \\\ \hline\end{array}$$

Use the following information to answer the next five exercises: A company wants to evaluate its attrition rate, in other words, how long new hires stay with the company. Over the years, they have established the following probability distribution. Let \(X=\) the number of years a new hire will stay with the company. Let \(P(x)=\) the probability that a new hire will stay with the company \(x\) years. What does the column \(" P(x) "\) sum to?
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