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Chapter 7: Problem 7

A box contains four slips of paper marked \(1,2,3\), and \(4 .\) Two slips are selected without replacement. List the possible values for each of the following random variables: a. \(x=\) sum of the two numbers b. \(y=\) difference between the first and second numbers c. \(z=\) number of slips selected that show an even number d. \(w=\) number of slips selected that show a 4

Short Answer

Expert verified
a. Possible values for x: 3, 4, 5, 6, 7\nb. Possible values for y: -3, -2, -1, 1, 2, 3\nc. Possible values for z: 0, 1, 2\nd. Possible values for w: 0, 1

Step by step solution

01

Define All Possible Pairs

Since we have 4 different numbers and we select 2 of them without replacement, the possible pairs of numbers are: (1,2), (1,3), (1,4), (2,1), (2,3), (2,4), (3,1), (3,2), (3,4), (4,1), (4,2) and (4,3)
02

Calculate Random Variable x

To calculate the value of the random variable x (The sum of the two numbers) for each possible pair, simply add the two numbers. This gives us: 3, 4, 5, 3, 5, 6, 4, 5, 7, 5, 6, 7. So the possible values for x are: 3, 4, 5, 6, 7
03

Calculate Random Variable y

To calculate the value of the random variable y (The difference between the first and second numbers), subtract the second number from the first. This gives us: -1, -2, -3, 1, 1, 2, 2, 1, 3, 3, 2, 1. So the possible values for y are: -3, -2, -1, 1, 2, 3
04

Calculate Random Variable z

The random variable z (The number of slips selected that show an even number) can be: 0, if neither of the two numbers is even; 1, if one of the two numbers is even; or 2, if both numbers are even. After examination of the pairs we find that the possible values for z are: 0, 1, 2
05

Calculate Random Variable w

The random variable w (The number of slips selected that show a 4) can be: 0, if neither of the two numbers is 4; or 1, if one of the two numbers is 4. After examination of the pairs we find that the possible values for w are: 0, 1

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Key Concepts

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

Combinatorics
Combinatorics is a branch of mathematics focused on counting, arranging, and combining objects. It's essential when dealing with problems where we need to determine the number of possible configurations, like picking slips from a box.

In the given exercise, we have a box with 4 slips of paper marked 1, 2, 3, and 4, and we're asked to select two slips without replacement.

Combinatorics helps us to systematically list all possible pairs that can be formed. We do not replace the slip once it's selected; this impacts the total number of combinations.
  • Each slip is unique, limiting total options.
  • Order matters since (1, 2) and (2, 1) are different pairs in this context.
  • The total possible pairs are calculated using permutations. Here, we have 12 combinations: (1,2), (1,3), (1,4), (2,1), (2,3), (2,4), (3,1), (3,2), (3,4), (4,1), (4,2), and (4,3).
Thus, combinatorics provides a structured way to tackle these problems and ensure all possibilities are considered.
Discrete Mathematics
Discrete mathematics involves study of integers and whole values, rather than continuous numbers. It's highly relevant in analyzing problems where distinct, separate values occur, like the numbers on the slips of paper.

This field plays a crucial role in this exercise as it helps in understanding and defining random variables, which can only take specific values.

Here, we determine the possible outcomes for various defined scenarios, like sums and differences of numbers on slips:
  • Variable x, representing the sum of two selected numbers, can only be specific integers between 3 and 7.
  • Variable y, the difference between the first and second selected numbers, has distinct possible values ranging from -3 to 3.
  • Variables z and w depict even number counts or selecting a slip marked 4, respecting discrete possibilities.
Discrete mathematics is all about such exact count and systematic analysis, significantly helping in classifying and exploring outcomes in a finite set.
Probability Theory
Probability theory provides the framework to quantify uncertainty, telling us how likely events are to occur. It's applied when examining the likelihood of drawing specific pairs of slips in this exercise.

While the exercise mainly revolves around random variable values, understanding how probability fits into this is crucial.

By defining random variables, we can examine:
  • The probabilities of outcomes like the sum of numbers (x) reaching a specific total, such as 4 or 7.
  • The chances of obtaining specific differences (y).
  • The likelihood of selecting even numbers (z) versus odds of grabbing a slip marked 4 (w).
Probability helps us determine how often these outcomes occur over many repetitions of the experiment, giving insight into expected outcomes when drawing from this finite sample set. This provides a deeper comprehension of chance beyond simple enumeration.

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Most popular questions from this chapter

You are to take a multiple-choice exam consisting of 100 questions with five possible responses to each question. Suppose that you have not studied and so must guess (select one of the five answers in a completely random fashion) on each question. Let \(x\) represent the number of correct responses on the test. a. What kind of probability distribution does \(x\) have? b. What is your expected score on the exam? (Hint: Your expected score is the mean value of the \(x\) distribution.) c. Compute the variance and standard deviation of \(x\). d. Based on your answers to Parts (b) and (c), is it likely that you would score over 50 on this exam? Explain the reasoning behind your answer.

Suppose a playlist on an MP3 music player consists of 100 songs, of which eight are by a particular artist. Suppose that songs are played by selecting a song at random (with replacement) from the playlist. The random variable \(x\) represents the number of songs played until a song by this artist is played. a. Explain why the probability distribution of \(x\) is not binomial. b. Find the following probabilities: i. \(p(4)\) ii. \(P(x \leq 4)\) iii. \(P(x>4)\) iv. \(P(x \geq 4)\) c. Interpret each of the probabilities in Part (b) and explain the difference between them.

Suppose that \(25 \%\) of the fire alarms in a large city are false alarms. Let \(x\) denote the number of false alarms in a random sample of 100 alarms. Give approximations to the following probabilities: a. \(P(20 \leq x \leq 30)\) b. \(\quad P(20

Determine each of the following areas under the standard normal \((z)\) curve: a. To the left of \(-1.28\) b. To the right of \(1.28\) c. Between \(-1\) and 2 d. To the right of 0 e. To the right of \(-5\) f. Between \(-1.6\) and \(2.5\) g. To the left of \(0.23\)

A breeder of show dogs is interested in the number of female puppies in a litter. If a birth is equally likely to result in a male or a female puppy, give the probability distribution of the variable \(x=\) number of female puppies in a litter of size 5 .
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