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Chapter 4: Problem 9

Express all probabilities as fractions. Your professor has just collected eight different statistics exams. If these exams are graded in random order, what is the probability that they are graded in alphabetical order of the students who took the exam?

Short Answer

Expert verified
\( \frac{1}{40320} \)

Step by step solution

01

Determine the total number of possible arrangements

To determine the total number of possible arrangements, calculate the factorial of the number of exams. Since there are 8 exams, we need to find the value of \(8!\).The formula for the factorial is given by: \[ n! = n \times (n - 1) \times (n - 2) \times \text{...} \times 1 \]So, for 8 exams:\[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \]Therefore, there are 40,320 possible ways to arrange the 8 exams.
02

Determine the number of favorable outcomes

The favorable outcome is the only specific arrangement where the exams are graded in alphabetical order. Hence, there is only 1 favorable arrangement.
03

Calculate the probability

Now, to find the probability that the exams are graded in alphabetical order, divide the number of favorable outcomes by the total number of possible arrangements.\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} = \frac{1}{40320} \]Thus, the probability is \( \frac{1}{40320} \).

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Key Concepts

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

factorial
In probability and combinatorics, the factorial is a key concept. It helps us determine the number of ways to arrange a set of objects. The factorial of a number is the product of all positive integers up to that number. It is denoted with an exclamation mark: For example: * \( n! = n \times (n-1) \times (n-2) \times \cdots \times 1 \) * For 8 exams, we calculate the factorial as \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \) * This results in 40,320 possible ways to arrange 8 exams The factorial function grows quickly. While 3! = 6 is still manageable, 10! = 3,628,800 shows how vast combinations can be as the number grows.
random arrangement
Random arrangement means organizing items without any specific pattern or order. In our problem, grading exams randomly could lead to any possible permutation of the exams being graded. Permutations are different ways to arrange a set of objects. With 8 exams, there are 40,320 possible permutations. Each permutation is equally likely if the exams are shuffled without any bias. Understand that with larger numbers, the permutations increase factorially. The concept of random arrangement is crucial in probability because it considers all possible ways to order the items, giving us the total outcomes for the scenario.
favorable outcomes
In probability, favorable outcomes are the specific results we are interested in. Among all possible outcomes, these are the ones that meet our criteria. For instance, grading exams in alphabetical order is a favorable outcome. Among 40,320 possible arrangements, only one of these sequences is alphabetical. To determine the probability of a favorable outcome, we use the formula: \( P = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \) In our case: \( P = \frac{1}{40320} \) This shows that the chances are very slim, but it quantifies the likelihood of our event happening.

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Most popular questions from this chapter

Answer the given questions that involve odds. In a clinical trial of 2103 subjects treated with Nasonex, 26 reported headaches. In a control group of 1671 subjects given a placebo, 22 reported headaches. Denoting the proportion of headaches in the treatment group by \(p_{t}\) and denoting the proportion of headaches in the control (placebo) group by \(p_{c}\) the relative risk is \(p_{t} / p_{c}\). The relative risk is a measure of the strength of the effect of the Nasonex treatment. Another such measure is the odds ratio, which is the ratio of the odds in favor of a headache for the treatment group to the odds in favor of a headache for the control (placebo) group, found by evaluating the following:$$\frac{p_{t} /\left(1-p_{t}\right)}{p_{c} /\left(1-p_{c}\right)}$$.The relative risk and odds ratios are commonly used in medicine and epidemiological studies. Find the relative risk and odds ratio for the headache data. What do the results suggest about the risk of a headache from the Nasonex treatment?

Find the probability.At Least One. In Exercises \(5-12,\) find the probability. Find the probability that when a couple has three children, at least one of them is a girl. (Assume that boys and girls are equally likely.)

Find the probability and answer the questions.MicroSort's XSORT gender selection technique was designed to increase the likelihood that a baby will be a girl. At one point before clinical trials of the XSORT gender selection technique were discontinued, 945 births consisted of 879 baby girls and 66 baby boys (based on data from the Genetics \& IVF Institute). Based on these results, what is the probability of a girl born to a couple using MicroSort's XSORT method? Does it appear that the technique is effective in increasing the likelihood that a baby will be a girl?

Use the given probability value to determine whether the sample results could easily occur by chance, then form a conclusion.A study addressed the issue of whether pregnant women can correctly predict the gender of their baby. Among 104 pregnant women, 57 correctly predicted the gender of their baby (based on data from "Are Women Carrying 'Basketballs'. by Perry, DiPietro, Constigan, Birth, Vol. 26, No. 3). If pregnant women have no such ability, there is a 0.327 probability of getting such sample results by chance. What do you conclude?

Find the probability and answer the questionsWhen Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 428 green peas and 152 yellow peas. Based on those results, estimate the probability of getting an offspring pea that is green. Is the result reasonably close to the expected value of \(3 / 4,\) as Mendel claimed?
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