Gibilisco S. Everyday Math Demystified: A Self-Teaching Guide

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this formula:

pðS [ D [ WÞ¼pðSÞþpðDÞþpðWÞ

 pðS \ DÞpðS \ WÞpðD \ WÞ

 pðS \ D \ WÞ

¼ 0:200 þ 0:100 þ 0:150

 0:030  0:110  0:020  0:010

¼ 0:280

This means that 280 of the students in the school are potential prospects.

Permutations and Combinations

In probability, it is often necessary to choose small sets from large ones, or to

ﬁgure out the number of ways in which certain sets of outcomes can take

place. Permutations and combinations are the two most common ways this

is done.

FACTORIAL

When working with multiple possibilities, it’s necessary to be familiar

with a function called the factorial. This function applies only to the natural

numbers. (It can be extended to more values, but then it is called the gamma

function.) The factorial of a number is indicated by writing an exclamation

point after it.

If n is a natural number and n  1, the value of n! is deﬁned as the product

of all natural numbers less than or equal to n:

n! ¼ 1  2  3  4  ... n

If n ¼ 0, then by convention, n! ¼ 1. The factorial is not deﬁned for negative

numbers.

It’s easy to see that as n increases, the value of n! goes up rapidly, and when

n reaches signiﬁcant values, the factorial skyrockets. There is a formula for

approximating n! when n is large:

n!  n

where e is a constant called the natural logarithm base, and is equal to

approximately 2.71828. The squiggly equals sign emphasizes the fact that

the value of n! using this formula is approximate, not exact.

CHAPTER 8 Taking Chances 181

PROBLEM 8-4

Write down the values of the factorial function for n ¼ 0 through n ¼ 15, in

order to illustrate just how fast this value ‘‘blows up.’’

SOLUTION 8-4

The results are shown in Table 8-3. It’s perfectly all right to use a

calculator here. It should be capable of displaying a lot of digits. Most per-

sonal computers have calculators that are good enough for this purpose.

Table 8-3 Values of n! for n ¼ 0 through n ¼ 15. This

table constitutes the solution to Problem 8-4.

Value of n Value of n!

424

5 120

6 720

7 5040

8 40,320

9 362,880

10 3,628,800

11 39,916,800

12 479,001,600

13 6,227,020,800

14 87,178,291,200

15 1,307,674,368,000

PART 2 Finding Unknowns

182

PROBLEM 8-5

Determine the approximate value of 100! using the formula given above.

SOLUTION 8-5

A calculator is not an option here; it is a requirement. You should use one

that has an e

(or natural exponential) function key. In case your calculator

does not have this key, the value of the exponential function can be found

by using the natural logarithm key and the inverse function key together.

It will also be necessary for the calculator to have an x

key (also

called x^y) that lets you ﬁnd the value of a number raised to its own

power. In addition, the calculator should be capable of displaying numbers

in scientiﬁc notation, also called power-of-10 notation. Most personal com-

puter calculators are adequate if they are set for scientiﬁc mode.

Using the above formula for n ¼ 100:

100! ð100

100

Þ=e

100

ð1:00  10

200

Þ=ð2:688117  10

 3:72  10

156

The numeral representing this number, if written out in full, would be a

string of digits too long to ﬁt on most text pages without taking up two

or more lines. Your calculator will probably display it as something

like 3.72eþ156 or 3.72 E 156. In these displays, the ‘‘e’’ or ‘‘E’’ does not

refer to the natural logarithm base. Instead, it means ‘‘times 10 raised to

the power of.’’

PERMUTATIONS

When working with problems in which items are taken from a larger set in

speciﬁc order, the idea of a permutation is useful. Suppose q and r are

both positive integers. Let q represent a set of items or objects taken r at a

time in a speciﬁc order. The possible number of permutations in this situation

is symbolized

and can be calculated as follows:

¼ q!=ðq  rÞ!

COMBINATIONS

Let q represent a set of items or objects taken r at a time in no particular

order, and where both q and r are positive integers. The possible number

CHAPTER 8 Taking Chances 183

of combinations in this situation is symbolized

and can be calculated as

follows:

=r! ¼ q!=½r!ðq  rÞ!

PROBLEM 8-6

How many permutations are there if you have 10 apples, taken 5 at a time in

a speciﬁc order?

SOLUTION 8-6

Use the above formula for permutations, plugging in q ¼ 10 and r ¼ 5:

¼ 10!=ð10  5Þ!

¼ 10!=5!

¼ 10  9  8  7  6

¼ 30ó240

PROBLEM 8-7

How many combinations are there if you have 10 apples, taken 5 at a time in

no particular order?

SOLUTION 8-7

Use the above formula for combinations, plugging in q ¼ 10 and r ¼ 5. We

can use the formula that derives combinations based on permutations,

because we already know from the previous problem that

¼ 30,240:

=5!

¼ 30ó240=120

¼ 252

Quiz

Refer to the text in this chapter if necessary. A good score is eight correct.

Answers are in the back of the book.

1. Empirical probability is based on

(a) observation or experimentation

(b) theoretical models only

PART 2 Finding Unknowns

184

(d) standard deviations

2. What is the number of possible combinations of 7 objects taken 3 at a

time?

(a) 10

(b) 21

(d) 210

3. What is the number of possible permutations of 7 objects taken 3 at a

time?

(a) 10

(b) 21

(d) 210

4. The diﬀerence between permutations and combinations lies in the

fact that

(a) permutations take order into account, but combinations do not

(b) combinations take order into account, but permutations do not

tions involve only discrete variables

(d) permutations involve only continuous variables, but combina-

tions involve only discrete variables

5. The result of an event is called

(a) an experiment

(b) a trial

(d) a variable

6. How many times as large is 1,000,000 factorial, compared with

999,999 factorial?

(a) 1,000,000 times as large.

(b) 999,999 times as large.

human-hours to calculate.

(d) There is not enough information given here to get any idea.

7. The set of all possible outcomes during the course of an experiment

is called

(a) a dependent variable

(b) a random variable

CHAPTER 8 Taking Chances 185

(d) a sample space

8. What is the mathematical probability that a coin, tossed 10 times in

a row, will come up ‘‘tails’’ on all 10 tosses?

(a) 1/10

(b) 1/64

(d) 1/2048

9. Two outcomes are mutually exclusive if and only if

(a) they are nondisjoint

(b) they have no elements in common

(d) they have identical sets of outcomes

10. The probability, expressed as a percentage, of a particular occurrence

can never be

(a) less than 100

(b) less than 0

(d) anything but a whole number

PART 2 Finding Unknowns

186

Test: Part 2

Do not refer to the text when taking this test. You may draw diagrams or use

a calculator if necessary. A good score is at least 30 correct. Answers are in

the back of the book. It’s best to have a friend check your score the ﬁrst time,

so you won’t memorize the answers if you want to take the test again.

1. Consider the following pair of equations, in which the intent is to ﬁnd

a solution or solutions, if any, for x and y simultaneously:

y ¼ 3x þ 4

y ¼ 3x  2

How many real-number solutions are there in this case?

(a) None.

(b) One.

(d) Inﬁnitely many.

(e) More information is needed in order to answer this question.

2. In the complex number 2 þ j3, the imaginary component consists of

(a) the number 2

187

(b) the number 3

(d) the number 3 multiplied by the square root of 1

(e) the product of 2 and 3, which is 6

3. In a frequency distribution, frequency is portrayed as the

(a) cumulative variable

(b) discrete variable

(d) random variable

(e) continuous variable

4. What is the mathematical probability that a coin, tossed ﬁve times in

a row, will come up ‘‘heads’’ on all ﬁve tosses?

(a) 1

(b) 1/2

(d) 1/25

(e) 1/32

5. Fill in the blank to make this sentence true: ‘‘A sample of a population

——————

of that population.’’

(a) an expansion

(b) a subset

(d) the average

(e) the median

6. In Fig. Test 2-1, suppose H

and H

represent two diﬀerent sets of out-

comes in an experiment. The light-shaded region, labeled H

, represents

(a) the set of outcomes belonging to H

but not to H

(b) the set of outcomes belonging to neither H

nor H

Fig. Test 2-1. Illustration for Part Two Test Questions 6 through 9.

PART 2 Finding Unknowns

188

or H

, but not both

(d) the set of outcomes belonging to either H

or H

, or both

(e) the set of outcomes belonging to H

but not to H

7. In Fig. Test 2-1, the dark-shaded region, labeled H

¼:H

, represents

(a) the set of outcomes common to both H

and H

(b) the set of outcomes belonging to neither H

nor H

or H

, but not both

(d) the set of outcomes belonging to either H

or H

, or both

(e) the set of outcomes belonging to H

but not to H

8. In Fig. Test 2-1, the entire portion that is shaded, either light or dark,

represents

(a) the set of outcomes common to both H

and H

(b) the set of outcomes belonging to neither H

nor H

or H

, but not both

(d) the empty set

(e) the set of outcomes belonging to H

but not to H

9. The Venn diagram of Fig. Test 2-1 portrays

(a) complementary outcomes

(b) mutually exclusive outcomes

(d) coincident outcomes

(e) nondisjoint outcomes

10. Which of the following is not an allowable maneuver to perform in

solving a single-variable equation?

(a) Adding a constant to each side.

(b) Subtracting a constant from each side.

(d) Multiplying each side by a variable that may attain a value of 0.

(e) All of the above (a), (b), (c), and (d) are allowable.

11. What is the solution of the equation 3x þ 5 ¼ 2x  4?

(a) There isn’t any.

(b) More information is needed in order to ﬁnd out.

(d) 9

(e) 9 or 9

12. What is the mathematical probability that an ‘‘unweighted’’ die, tossed

three times, will show the face with 2 dots on all three occasions?

(a) 1 in 6

Test: Part 2 189

(b) 1 in 36

(d) 1 in 216

(e) 1 in 46,656

13. What is the solution set of the equation x

þ 64 ¼ 0?

(a) The empty set.

(b) {8}

(d) { j8, j8}

(e) More information is needed to answer this.

14. Fill in the blank to make the following sentence true: ‘‘If the number

of elements in a distribution is odd, then the

————————

is the

value such that the number of elements having values greater than

or equal to it is the same as the number of elements having values

less than or equal to it.’’

(a) mean

(b) deviation

(d) mode

(e) median

15. Let q represent a set of items or objects taken r at a time in a speciﬁc

order. The possible number of permutations in this situation is

symbolized

and can be calculated as follows:

¼ q!=ðq  rÞ!

Given this information, what is the possible number of permutations

of 8 objects taken 2 at a time?

(a) 40,320

(b) 720

(d) 28

(e) 24

16. Let q represent a set of items or objects taken r at a time in no

particular order, where q  r, and where both q and r are positive

integers. The possible number of combinations in this situation is

symbolized

and can be calculated as follows:

¼ q!=½r!ðq  rÞ!

PART 2 Finding Unknowns

190