Math: Expected Value

Tutoring math, you might be asked about statistics, in which expected value is an early topic.

In many everyday situations, the expected value is equal to the mean, aka the average.  The difference is more in definition than in practice.  Expected value involves probability, whereas mean involves fact.

Formally, the expected value is defined as follows:

Expected Value=Σoutcome_value*probability

The Σ symbol means “sum”.  In other words, the expected value takes each possible outcome and multiplies it by its probability.  Then, it adds all those products together.

Example 1: Give the expected value of rolling a fair six-sided die.

Solution:  We know that the probability of getting each result is 1/6.

The expected value, E, of the die roll is

E=(1/6)1+(1/6)2+(1/6)3+(1/6)4+(1/6)5+(1/6)6=(1/6)(1+2+…+6)=21/6=3.5

Notice that the expected value is not necessarily a possible value. Not surprising, really: if you take the average height of ten people, you’ll likely arrive at a height that none of them is.

Example 2: Find your expected payoff in the following situation: 1000 people pay $1 each for a ticket. Then a number is drawn. The holder of the winning ticket gets $500.

Solution: The probability of winning is 1/1000, in which case you get $500. Really though, the payoff is only $499, since you had to spend $1 to buy the ticket. The probability of not winning is 999/1000, in which case you get nothing. Once again, you still had to pay $1 to lose, so the payoff is -$1. Applying our definition of expected value we get

E=(999/1000)(-1) +(1/1000)(499)=-0.999+0.499=-0.50

So, your expected payoff is -$0.50. It had to be a loss, since not all the money paid in was awarded.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Exponential Growth: an interesting application

Tutoring math 12, exponential growth is “always on my mind.”

Years ago, I used to read the Economist.  Eventually I became too busy to keep up with it, but I always enjoyed it when I could.

One of the last articles I remember (this was maybe in ’05 or ’06), China’s economy was being compared with India’s.  At the time, China’s growth was 8%, while India’s was 6%. Either rate would signify wildfire growth in a developed economy; I’d say Canada will be lucky to grow at 2% this year. However, the article said that India’s growth, while a very nice 6%, melted in comparison with China’s 8%.

As a math tutor, I thought about that comment for a moment. “Is 8% really that much more than 6% growth?” I asked myself.

The key is that it’s exponential growth. This year’s growth becomes a part of next year’s economy, which then grows again, so you get growth on growth on growth. That’s exponential growth: anything natural grows that way. My earlier article here talks more about it.

Reading that earlier article, you’ll also encounter the law of 72, which states the following about an economic entity:

(growth rate)x(doubling time)=72.

It’s an approximation, but a very good one.

Let’s compare India’s historic growth at 6% with China’s at 8% using the law of 72. Does 8% really “melt” the 6%? Well, what we can say is that, by the law of 72, India’s economy will double every 12 years, while China’s will double every 9 years. For simplicity, let’s imagine the economies begin at the same size. In 36 years, India’s will double three times (every 12 years), so it will be 8 times its original size. (2x2x2=8). In that same period of 36 years, China’s will double four times (every 9 years), reaching 16 times original size. (2x2x2x2=16). If they were the same size at the beginning, China’s economy, having doubled an extra time, is exactly twice India’s at the end of the 36 years.
 
From that point of view, the difference in the growth rates is impressive.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: inflation and interest

Recalling the 70s and 80s, the math tutor proposes a nostalgic premise for your coffee break.

Inflation eats away at your money.  Right now, inflation is very low; it has been since the late 90s. However, in the 70s and 80s inflation was significant – and very high some years.

You can tell inflation is at work if prices are going up.  I remember chocolate bars, chips, and pop used to cost a quarter each when I was six.  A year later, they all cost 30 cents.

To calculate the inflation rate in the case above, we do this:

rate=(change in price)/(former price)

In our case

rate=5/25=0.20

To change from a decimal to a percent we move the decimal point two jumps right:

0.20=20%

Of course, 20% inflation is very high; I don’t imagine inflation is even 2% right now.

From what I’ve read, interest rates generally run around 3% above inflation. That’s a historical trend, so it may not be the case at any given time. However, if interest offered to savers is 6%, inflation is likely around 3%.

Let’s find the change in value of $1000 over a year at 6% interest with 3% inflation (we assume the $1000 is in a savings account):

Interest=principal*rate

The principal is the amount deposited in the account. Of course, the rate must be in decimal form. To go from percent to decimal, you shift the decimal point two places left (or else you can just divide by 100):

6%=0.06

Now we can find the interest earned:

Interest=1000*0.06=60

Before considering inflation, the $1000 has grown to $1060.

Now let’s witness the action of inflation. The real value of the $1060 at year end is as follows:

real value=(1-inflation rate)*dollar_amount

Once again, the inflation rate must be in decimal form. We realize that 3% is 0.03 and proceed:

real value=(1-0.03)*1060=0.97*1060=1028.20

So the proceeds are $28.20 after inflation.

In a future post I’ll mention a surprising twist about the relationship between inflation and interest:)

 
Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: Factoring Easy Trinomials

As a math tutor, you teach and review this method constantly.

Back in May, I began a series of posts about factoring polynomials.  To refresh the topic, you can check here, here, and here.

Factoring polynomials is a make-or-break skill for high school students taking academic math. It encompasses about five techniques, of which easy trinomial factoring is probably the best known. Let’s have a quick look:

Example 1: Factor x2 -3x -28

Solution: Since the coefficient of x2 is 1 (which we know because there is no number written in front of it), we can use the easy trinomial method.

Step 1: Write (x      )(x     )

Step 2: After the x’s, write the numbers that will multiply to give -28, but add to give -3.

You have to do some mental math: 7×4=28, but one of the numbers has to be negative to give -28. The numbers must be -7 and +4, since -7+4=-3.

(x -7)(x +4)

The answer is (x – 7)(x + 4). You can verify using the foil method:

First: x*x=x2

Outer: x*4=4x

Inner: -7*x=-7x

Last: -7*4=-28 (remember: negative times positive gives negative)

Now, line up the four terms we just obtained:

x2 +4x -7x -28

We can combine the like terms: 4x – 7x = -3x

Finally we get

x2 -3x -28.

If you foil out your answer and get back the original trinomial, you know it’s right.

Example 2: Factor x2 + 5x + 4

Solution: The numbers that multiply to give 4 but add to give 5 are 1 and 4: 1*4=4, 1+4=5.

Therefore, the answer is (x + 4)(x + 1)

Example 3: Factor x2 -10x + 16

Solution: The numbers that multiply to give 16 but add to give -10 are -8 and -2 (recall that negative times negative gives positive): -8*-2=16, -8+-2=-10

The answer is (x – 8)(x – 2)

Example 4: Factor x2 +5x – 14

The answer is (x + 7)(x – 2)

Good luck with this method. Most people like it once they get used to it:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

French education: the challenge of exogamy

In BC, 75% of francophone families are exogamous.

Exogamy refers to the marriage of someone from a certain culture, to a spouse from outside that culture.  From the francophone perspective, an exogamous family has one parent with French as the mother tongue, while the other parent has a different mother tongue.

In the francophone education system, most students come from exogamous families. As homogeneous French families become increasingly rare in Canada, the survival of francophone education outside Quebec depends on the enrollment of children from exogamous families.

Many people wonder why they would send their child to francophone education when they could just send them to the English system.  The answer is that in Canada, children who are educated in French usually turn out to be better in English as well.  Most people accept without question that knowing a second language is advantageous, and that learning it from a young age – if possible – is the best way.

Surprisingly, a francophone parent will often speak English at home to their children.  At the same time, the exogamous parent (usually English-speaking) may be more serious about their children’s learning French – probably because it’s a great opportunity that the English parent never had themselves.

The challenge for the francophone schools is to devise a way to welcome the non-French parents of exogamous families, while still maintaining a French-speaking environment.  Such a solution will likely ensure the growth of French-English bilingualism outside Quebec.

Sources:

Rodrigue Landry, “The challenges of exogamy”

“English Information,” Conseil scolaire francophone de la Colombie Britannique

Math: solving percents with cross-multiplication

As a math tutor, you encounter this topic often – especially with students in vocational training.

Many ways exist to tackle percents.  However, the advantage of the cross multiplication method is its usefulness in virtually any situation involving them. If you missed my blog entry introducing the cross multiplication method, find it here.

Example 1: What is 15% of 390?

The key with percents is to realize that, for instance, 15% means 15 over 100. That is,

15%=15/100

We can now incorporate our cross mutliplication scheme:

15/100 = x/390

Of course, x represents the number we need to know.

Now we do the actual cross-multiplication:

15*390=100x

Dividing out 100 from both sides, we get

15*390/100 = x

We discover that 15% of 390 is 58.5.

Some other uses of cross multiplication with percents will be covered in future posts. For now, go back to the sun:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

French: practical hints for typing accents

When you tutor French, a student might ask how to produce the accents on an English keyboard.  Here are a couple of options:

On a Microsoft product, every French accent has its own Alt+(4 digit code).  For instance, this ç was typed using Alt+0231.  You need to hold down the Alt key, then type 0231 using the numeric keypad to the right – not the numbers across the top. Here are some codes that, once memorized, can really speed up your French typing:

é:     Alt+0233

è:     Alt+0232

à:     Alt+0224

ç:     Alt+0231

î:     Alt+0238

ô:    Alt+0244

Another way to produce accents is to use the character map.  Look under All Programs→Accessories→System Tools and you should see it.  It’s a grid of different characters which you can copy and paste to your work – really a great tool.

If you’re in Word, of course, you can go Insert→Symbol to find everything you need.  Word has its own shortcut sequences; my wife uses them all the time.  However, the sequences above will work in Word as well.

If you like the copy and paste method, here’s a time saver:  Copy and paste all the accents you’ll need when you first start, like so:

à  ç  è  é  î  ô

Now, you don’t have to flip back and forth between a menu and your work; you can just copy and paste from the list.

I hope this helps:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: What is a “perfect number?”

While enjoying this surprisingly fine summer, the math tutor recalls a definition.

When I was a kid, I read in one of my math texts (it might have been grade six) that a “perfect number” is one whose factors (except itself, of course) sum to it.

Example:  6 is a perfect number.

Factors of 6:    1, 2, 3, 6

Sum of factors (excluding 6 itself):  1+2+3=6

Example:  28 is a perfect number.

Factors of 28:   1, 2, 4, 7, 14, 28

Sum of factors (excluding 28 itself):   1+2+4+7+14=28

I always thought it was an interesting definition.  Curiosities like this are great; they keep you thinking about math when you don’t have to do any:)

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Math: What is a Radian?

When you tutor math, you explain radians every semester to your grade 12 students.

Most people begin measuring angles in degrees.  However, you can also measure an angle in radians.  1 rad≈57.3°.

While degrees come from (I am told) Babylonia, or one of the ancient civilizations of that area, radians are a “natural” way to measure angles.  Behold:


In the above picture, CA is a radius. The arc from A to B is the same length as CA. Therefore, angle ACB is 1 radian. 1 radian is the angle that you traverse by following an arc the length of the radius. Said another way, it’s the angle subtended by an arc one radius long.

Recall that the circumference of the circle is 2πr, where r is the radius.  Since 2πr is the exact circumference, 2π radians is exactly 360°.

Radians can be referred to as rads, but are usually stated without any unit. That’s how you can tell which way the angle is measured:  if it’s in degrees, it will have a degree sign.  If it’s in rads, it won’t have any units.  Therefore, an angle of 54° means, of course, 54 degrees.  However, an angle of 32 means 32 rads.

Please keep enjoying this fine summer!

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.

Practical math: Some easy conversions from metric to imperial

As a math tutor, you realize that conversions from metric to imperial are part of the grade 10 curriculum.  Let’s talk about a few that don’t need a calculator.

Even though the (Canadian) high school student grows up in a metric environment, the trades use both systems.  Moreover, the tutor likely grew up in the 70s, so still thinks as much in imperial as metric.

With a calculator, of course, you can easily convert any measurement to any other. Nowadays, you can just key a measurement into your browser and it will return the conversion.  In summer, however, such questions seem to arrive more often in everyday life – possibly when you’re not at your computer.

So, both for those in summer school, as well as those who might find these tricks useful in every day life, here are some simple conversions you can do in your head. While not exact (I think the temp conversion is), they get you within 2% of the answer.

kg to pounds:  double it, then add 10% of the answer.

example:  77 kg to pounds

step 1:  double the mass in kg:  77 times 2 = 154.

step 2:  add 10% more.  15.4 + 154= 169.4

So, 77kg is 169.4lbs.

metres to yards:  just add 10%.

55m is 55 + 5.5 or 60.5 yards.

inches to cm:  multiply by 5, then divide by 2.

4 inches = 5(4)÷2 = 10cm.

Fahrenheit to Celsius:

This conversion comes up a lot, but there is no convenient way without a calculator. You subtract 32 from the Fahrenheit, then divide by 1.8.

Example:  Convert 80F to C

step 1:  80-32=48

step 2:  48÷1.8=27 (rounded to the nearest whole degree).

So, 80F is 27C.

Here’s an irony about summer measurements:  According to Wikipedia, the Canadian football field is 110 yards, whereas the American is 100.  However, the Canadian football field is 100 m (since going from metres to yards you just add 10%).  So the American and Canadian are both 100 long in their own units.

Jack of Oracle Tutoring by Jack and Diane, Campbell River, BC.