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Chapter 4: Problem 21

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

Short Answer

Expert verified
The total probability should be 1, not 1.05. Adjust \(P(5)\) from 0.15 to 0.10 to correct this.

Step by step solution

01

Identify the Probability Distribution Condition

For any probability distribution, the sum of all the probabilities must be equal to 1. Let's add the given probabilities: \( P(1) = 0.15 \), \( P(2) = 0.25 \), \( P(3) = 0.30 \), \( P(4) = 0.20 \), \( P(5) = 0.15 \). We will calculate their total.
02

Calculate Total Probability

Add all the probabilities together: \(0.15 + 0.25 + 0.30 + 0.20 + 0.15 = 1.05\). Since the total probability is 1.05, it indicates a mistake because it should be exactly 1.
03

Verify Each Row Calculation

Verify each row in the table to ensure multiplication accuracy for \(x \times P(x)\):- Row 1: \(1 \times 0.15 = 0.15\)- Row 2: \(2 \times 0.25 = 0.50\)- Row 3: \(3 \times 0.30 = 0.90\)- Row 4: \(4 \times 0.20 = 0.80\)- Row 5: \(5 \times 0.15 = 0.75\)
04

Identify and Correct Probability Error

Given all row calculations are correct, the mistake is in the initial probability entries. Possible corrections could involve adjusting one or more probabilities such that their sum is exactly 1. For instance, reducing \( P(5) \) from \( 0.15 \) to \( 0.10 \) would make the total sum 1.00.

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Key Concepts

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

Understanding Probability Sum
When working with probability distributions, one of the most important rules is ensuring that the sum of all probabilities equals 1.
This represents the fact that, collectively, all possible outcomes exhaust the certainty of the situation.
For instance, when rolling a six-sided die, each side has an equal probability of landing face-up and if you add these probabilities together, they will total to 1.
To check for any mistakes in a probability table, this rules becomes handy.
You simply add up the probabilities and see if they equal 1.
If they do not, you know there is an error.
In our exercise, the sum of the probabilities was 1.05, indicating a mistake has occurred.
This Discrepancy suggests that some probabilities might be incorrectly listed or need adjusting to properly total to the correct value.
Error Identification in Probability Distribution
Finding the error in a probability distribution involves several steps.
After realizing the sum of probabilities is incorrect, a thorough check of individual values is necessary.
  • First, double-check the original probabilities.

  • Ensure there were no typos or miscalculations when they were first written.

  • Consider the logical fit of each probability 鈥� for instance, in our problem, each number should logically add to a whole sum of certainty, 1.

Once you identify which number (or numbers) are incorrect, you can start adjusting.
Reducing or increasing probabilities to ensure they correctly add up without losing the validity of the scenario is key.
In our example, one adjustment was reducing the last probability from 0.15 to 0.10, which fixes the sum error and retains the integrity of the probability distribution.
Checking Table Calculation Accuracy
Ensuring the accuracy of step-by-step calculations in tables is crucial in statistics.
Each row of a probability table should be verified for correct multiplication of the outcome (x) by its probability, P(x).
  • For example, if x is 3 and P(x) is 0.30, the resulting calculation should be 0.90.

  • Carefully checking each line prevents gradual errors that could significantly distort the final results.

  • In our exercise, every row's product of x and P(x) was correctly calculated, confirming that these are not contributing to the sum error.

This verification ensures that there are no arithmetic mistakes at the line-item level.
While it didn't reveal the source of the problem with the sum in our scenario, it's an essential step in verifying accuracy in any statistical problem.
Always recheck table calculations when encountering unexpected results 鈥� it might sometimes reveal where things went wrong.

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Most popular questions from this chapter

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. More than 96 percent of the very largest colleges and universities (more than 15,000 total enrollments) have some online offerings. Suppose you randomly pick 13 such institutions. We are interested in the number that offer distance learning courses. 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. On average, how many schools would you expect to offer such courses? e. Find the probability that at most ten offer such courses. f. Is it more likely that 12 or that 13 will offer such courses? Use numbers to justify your answer numerically and answer in a complete sentence.

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. The literacy rate for a nation measures the proportion of people age 15 and over that can read and write. The literacy rate in Afghanistan is 28.1%. Suppose you choose 15 people in Afghanistan at random. Let \(X =\) the number of people who are literate. a. Sketch a graph of the probability distribution of \(X\). b. Using the formulas, calculate the (i) mean and (ii) standard deviation of \(X.\) c. Find the probability that more than five people in the sample are literate. Is it is more likely that three people or four people are literate.

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?

Use the following information to answer the next six exercises: The Higher Education Research Institute at UCLA collected data from 203,967 incoming first-time, full-time freshmen from 270 four-year colleges and universities in the U.S. 71.3% of those students replied that, yes, they believe that same-sex couples should have the right to legal marital status. Suppose that you randomly select freshman from the study until you find one who replies 鈥測es.鈥� You are interested in the number of freshmen you must ask. Construct the probability distribution function (PDF). Stop at \(x = 6\). $$\begin{array}{|c|c|}\hline x & {P(x)} \\ \hline 1 & {} \\ \hline 2 & {} \\\ \hline 3 & {} \\ \hline 4 & {} \\ \hline 5 & {} \\ \hline x & {P(x)} \\\ \hline 6 & {} \\ \hline\end{array}$$

A consumer looking to buy a used red Miata car will call dealerships until she finds a dealership that carries the car. She estimates the probability that any independent dealership will have the car will be 28%. We are interested in the number of dealerships she must call. 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. On average, how many dealerships would we expect her to have to call until she finds one that has the car? e. Find the probability that she must call at most four dealerships. f. Find the probability that she must call three or four dealerships.
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