Objective: To demonstrate the techniques used to find the probability of winning a lottery, and to illustrate just how small the chances of winning a typical lottery are. In the course of explaining the necessary counting techniques, we will introduce the Fundamental Counting Principal, factorials, permutations, combinations, probability, and odds.
Hi, and welcome to this week’s episode of Math Matters. I’ve got some neat stuff to show you today, some stuff that’s kind of important, especially in the state of Kentucky. And then some other things that involve some investment strategies and just general kind of getting-by strategies. I want to get started on a little feature on probability and mostly I’m going to talk about counting. I know you thing well, one, two, three, four, five that’s counting, but that’s kind of what I’m talking about, really what I’m saying is counting how many ways certain things can happen and then how that affects – we’ll end up talking about probability – before we’re done. We are in a lottery state and I think it’s important for folks to understand some of the percentages, the chances of winning these things, and I’m going to take you through that math this morning for how you calculate those odds of winning and those kinds of things.
Now here in Kentucky, our big, our big lottery in the state of Kentucky is Lotto South and for a dollar you get to pick six numbers one through 49 with no repeats. You can get them in any order, you win. Now, what we’re going to do is learn how to count the number of possible tickets. And that’s all we have to do. They sell like a million of these things, but if two people have the same winning ticket, they have to share the, share the pot. So really what matters is what are the odds, what are the chances of picking that winning ticket.
(Figure 1)
And so that’s what I’m going to take you through now and it all kind of starts with this idea called the fundamental counting principle. Which just says this: the number of ways to place items into categories is the product of the number of choices per category. And product means we’re going to multiply, so let me give you a little example. Let’s say Bill had five different shirts, he had six different pair of jeans, and ten different pair of socks and they’re all compatible. They all, you know, match some way or another. How many different outfits are possible? All we do is we’ve got three categories: shirts, jeans, socks, and we put in the number of choices for each: five, six, and ten, and you multiply and so he’s got 300 potential outfits.
(Figure 2)
Now I can make things slightly different if I say all the choices are kind of being removed from the same pot and that makes, that gives me a chance to make use of some nice notation.
Let me introduce this notation. This is called “n” factorial. It looks like “N” (loudly and excitedly,) because it’s got that big explanation mark, but it just means you take “n” and you multiply it by one less and one less and one less until you get down to one. And we define
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Figure 1, Segment 1
Figure 2, Segment 1
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zero factorial to be one. That’s a notational thing. Hopefully I’ll get to show you why we do that in just a moment. So if I look at this situation, let’s say I had five runners and I say well how many different ways can they finish the race? Well, I’ve got five categories and five choices for the first one and the choices go down, four, and then three, and then two, and then one. And see, that’s five factorial which equals 120 which would be 120 ways that could happen. In general, that means that there are “n” factorial ways to arrange “n” things.
(Figure 3)
Figure 3, Segment 1
Okay, now let me change the situation slightly. Let’s look at a situation where I’ve got 20 contestants and I want to say how many ways can they finish first, second, and third? Well it’s not three factorial and it’s not 20 factorial. I’ve got three categories. I’ve 20 choices for the first one, 19 for the second, 18 for the third and that equals 6,840. But I can use that factorial notation if you notice I can write those and then have a 17 factorial in the top and the bottom, that’s just one, and then I can rewrite this as 20 factorial and then, 20 minus three, over 17 factorial. And that gives me a general way to “r” of “n” things where “r” is no bigger than “n”, that’s kind of this fraction right here.
(Figure 4)
Everything I’ve done up until this point has had order involved, now I want to make the order of things fall out of it. So let me give you another situation. Let’s say there’s 30 people at a reception and they decide they’re going to shake hands with every other people in the room. So, how many handshakes is that? Well, if we had a first hand-shaker and a second hand-shaker, it’d be 30 times 29. That’s the fundamental counting principle, but there isn’t really any difference between Bill shaking hands with Sue and Sue shaking hands with Bill.
And we have counted those occurrences then twice. So what we do then is look at the number
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Figure 4, Segment 1 of different arrangements of the two things, okay, and then we divide by that so that gives us, go back and flash the total here, there is a correct answer of 435 hand-shakes. And so that gives us then a general kind of formula then for how to arrange “r”, or not to arrange, but how to group “r” of “n” things where “r” is no bigger than “n”. This is our general formula that we had and then “r” is what we divide because there are “r” factorial ways to arrange
“n” things, kind of divide out the order in other words.
(Figure 5)
Okay, now I’m ready to address the Lotto problem: how many different lottery tickets are possible? We’ve got 49 different numbers to choose from, we’re going to choose six and it doesn’t matter what order they fall. Okay, I’m going to use this formula: I’ve got 49 numbers to choose from, I’ve got, and to get them arranged I would say 49 minus six, that’s the factorial and then to get rid of the order, I divide by six factorial. Okay, then you just have to do the arithmetic on that which sounds horrible, but it’s really not. You can expand
49 factorial into 49, 48, 47, 46,45,45,44, and then stop at 43 factorial and cancel those big buggers out. Those are large numbers, but we can get rid of them pretty quick, the rest of this is just a division problem, which you can do on your calculator. Turns out that that is
13,983,816. That’s how many different lottery tickets are possible.
(Figure 6)
Now I want to look at probabilities for a second place chance. I want to talk about chance in general because that’s really what we’re working for. What’s the chances, what are the chances that I would win the lottery? We can talk about this in two ways: we can talk about probability, which is the number of successful outcomes over the number of possible outcomes or we can talk about odds, which is the ratio of successful outcomes to unsuccessful outcomes.
Okay, now they are very close to each other. We typically we see odds expressed like this
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Figure 5, Segment 1
Figure 6, Segment 1
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with a colon between them. If you know one, then you can calculate the other, bit I’m not going to get into that. I mean that’s something we can do at a later date if we want to. I’m going to stick to odds because we’re playing a game and typically gaming things are reported in odds.
(Figure 7)
Figure 7, Segment 1
So, if we look at the number of possible tickets here, one of those is a winning number, the others will not be the winning number, so the odds of winning Lotto South would be one to 13,983,815, one less than this. Actually on the Kentucky Lottery page it says, it compares this to 13,983,816. What they did is they calculated the probability and then reported it as odds, which is technically incorrect, but I’ll give them a pass on that.
(Figure 8)
And just out of curiosity, I went to the national weather service and looked up the odds of getting struck by lightning. This is something you need to know if you’re going to play the game in Kentucky. The odds of getting struck by lightning this year would be one in 700,000.
That number is half the number that we just calculated which means you’re twice as likely to get struck by lightning as you would be to win the Lotto South in one play. That’s stuff
I think you need to know if you’re going to be playing the lottery in the state of Kentucky.
(Figure 8)
Excuse my lousy arithmetic – 700,000 is roughly
1 the number we just calculated. That
20 means that if we assume that the probability of getting struck by lightning is uniformly distributed over the entire year (365 days,) the probability of winning the Lotto South with one ticket purchase is roughly equivalent to the probability of getting struck by lightning during any
365 ≈ 18 day period.
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I covered a lot of stuff, so I’m going to flash up the important stuff on some summary
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Figure 8, Segment 1 pages here and let you kind of digest that for a few moments.
Summary page 1, Segment 1
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Summary page 2, Segment 1
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Objective: To illustrate the relevance and the underlying mathematics of being able to convert from one unit of measure to another.
Another thing I’d like to talk to you today about is a bit on unit conversion, and what
I mean here is that we’re given amounts in different units of measure. In order to do the mathematics involving units of measure, you really have to get everything to kind of the same unit and that is a crucial fundamental skill for someone who is really going to apply mathematics. You have to be able to do this. This is one of the first things you have to do.
And it’s just a little touch of algebra and really just fractional arithmetic and I’d like to show you that, how to do that.
I do want to say this about it. It does seem that this has been taught in a fairly convoluted way. We teach you if you’re going from small units or big units to small units you do one thing and then if you’re going from small units to big units, you do the other. And I think that’s kind of confusing, I think there’s a better approach to this and I’m going to show you a more uniform method for doing this, where you do the same thing every time.
(Figure 1)
Okay, it’s guaranteed to work. I call it unit cancellation. Some text books, particularly math, elementary ed for math teachers text books will call this dimensional analysis, but that just sounds like something off of Star Trek to me. It just makes it sound scary. I just want to multiply by one: that’s it, just multiply by one and then the one that we use depends on the conversion rates, okay?
(Figure 2)
We have to know the conversion rates if we’re going to convert from miles to feet or feet to miles, we have to know that one mile is 5,280 feet. But that gives me two ways to write one. I can write one mile over 5,280 feet, those are the same amounts so that’s one, or I could reverse it – 5,280 feet over one mile, that’s one. And then what you do is you arrange the units you’re trying to get rid of in the numerators and the denominators opposite of each other so that we can just cancel them out. We just treat them like a factor in the problem; we just get rid of them.
(Figure 2)
I’ll give you an example. Let’s say I want to convert 0.6 miles to feet. So I take my 0.6
miles and I actually I think of 0.6 as
6
, so 6 in the numerator, 10 in the denominator, and
10 then I put the unit in the numerator and then I multiply by one. And I do it in such a way that the mile unit here and the mile unit here can be cancelled. That’s a slick little trick and then I just multiply things out. You can do some simplifying if you want to, but that works out to 3,168 feet, and yes I did go from big to small and I multiplied, but that takes care of it for me.
(Figure 3)
Let me show you how to do little to big. For example, I want to convert 40 inches to yards. Well, 40 inches, I’ll put that in my numerator and then I need to know the conversion between inches and yards, but I can go part of the way. I know that 12 inches is one foot so
I can get that cancelled out and now I just need to convert from feet so that I know that three feet is one yard, cancel those out and then I just work it out – ten ninths or if you prefer one with a repeating one tenth. And yes, I did go from little to big and yes I did divide, but see that system does not confuse.
(Figure 3)
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Figure 1, Segment 2
Figure 2, Segment 2
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Figure 3, Segment 2
What if there are square units are in the numerator and in the denominator? Okay, well let’s look at an example like that. Say we wanted to convert two square miles to square kilometers. Well, I put the two square miles in there. I have to look up, I don’t know this one off the top of my head, I had to look it up, but one mile is equivalent to roughly to one point six zero nine three kilometers. I can cancel the miles once and then to cancel them a second time think of this as squared, you know times itself, so I’ll just multiply by it again and then
I can cancel, cancel, cancel. And in the end result, it gives me kilometers squared. I just have to work the problem out; it’s about five, a little over five square kilometers.
(Figure 4)
Here’s another one and this is where it gets a little sticky. Say I was going to go from something like miles per hour to feet per second. Now not only do I have to change miles to feet but I have to change hours to seconds. Okay, again multiply by one, it’s just this easy. Just arrange things so that you can cancel the units. One mile per hour is one in the numerator and then hour in the denominator. I’ve already talked about how one mile is this many feet and I arrange it so that I can cancel the miles. Now let’s get down to the hours. One hour, well 60 minutes, maybe you don’t know how many seconds are in an hour.
So one hour is 60 minutes and then you can convert one minute to 60 seconds and you’re just multiplying by one each time. You’re not changing the value when you multiply by one.
You get all of this done, you do the arithmetic and it works out to
22 feet per second or if
15 you prefer decimals, that’s almost one and a half feet per second. Okay, the bar over the six indicates that that is a repeating six, that goes on forever. And in that case you’ll notice that
I both multiplied and divided. That’s a situation where what they’ve taught in algebra doesn’t come out very much.
(Figure 4)
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Figure 4, Segment 2
Now you may be saying “Well, you know, that’s great, but how does that help me out?
You know everyday stuff, I’m not concerned with serious math problems.” Let me give you a situation where this would be helpful. Let’s say that you’ve just had your ride tricked out. You’ve got a bigger set of tires on it now than you originally had and although your speedometer says 65 miles per hour, you’re actually driving faster than that. You don’t want to get any tickets because you’re out of points or whatever and so how can you, how can you actually check to see how fast you’re going? Well, try this little test. Get on an interstate or parkway where you can safely drive 65 and not have to slow down or speed up. Set your cruise control to that speed. It could be actually any speed, but I’m worried about 65 because after that’s where I get tickets. And then, it’s important here that you go a constant rate of speed throughout this and then drive between two consecutive mile markers and then time yourself. See how many seconds it takes you to get from one mile marker to the next. So let’s say it takes, takes me 50 seconds to drive between two mile markers, okay? So what
I have done is I have driven one mile in 50 seconds. Let’s convert that to miles per hour.
(Figure 5)
The seconds, I would convert to, well I would do the conversion here, one hour and I already learned this it’s 36 hundred seconds. The seconds cancel out and that works out to be 72 miles per hour. That’s an indication if you were driving a constant 65 that you ended up with a 72, you’re actually driving 72 miles per hour. You need to work on getting that speedometer adjusted or at least just learn what is 65 miles per hour on your speedometer.
(Figure 5)
I actually flashed up a few things here that might take a little time to digest, so, again,
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Figure 5, Segment 2 before we go to the last segment, I’m going to flash up some summary pages that kind summarize everything up until this point.
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Summary page, Segment 2
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Objective: To illustrate how radicals and radical expressions can be used in a real-world application, specifically calculating the average growth rate of an amount receiving compound interest.
We show that the arithmetic mean is inadequate for this, and introduce the geometric mean as tool for finding the correct answer in this situation.
The last thing I want to talk to you about today is radical expressions, expressions that have a radical sign in them and this is something when it is presented in algebra class that is a little bit dry and I’ve even had, this is a little discouraging, I’ve even had algebra teachers out in the public school system say “You know, what’s a good use for these things? Why do students need to know these things?” So I’m going to show you a real nice application of radical expressions today and it all has to do with finding average growth rates.
Okay, let me describe a situation to you where you’ve inherited some stocks from a rich relative and the shares are valued at $ 0.32 each at the time that you get them, but eight years later, they’re valued at a hundred and fifty, I’m sorry, $ 1.52 each. So you’re trying to kind of plot their average rate of growth here so you can determine how much money later on you’ll have later on when you retire. And you may even have a financial planner looking at these ideas so the question is “How do you calculate the average rate of change, the average rate of growth of these shares?” (Figure 1)
Figure 1, Segment 3
So we could just simply do a calculation and divide by eight, I mean the percent growth is where you take the difference between the new and the old and then you divide by the old,
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that’s percentage growth up or down. And really you can think of that as new divided by old and then just subtract the 100%. So in this situation, here’s the new price, there’s the old price and then you subtract the one after you’re done dividing. That gives you 3.75, which means the stock has grown total 375%. Now I guess you could say “Well, does that mean then the annual growth is, you know, take that, divide by eight, is that nearly 50% growth each year?” 46.875%? Well, the way to test this, we talked about how to do this on a previous episode, is you say okay, you take your previous growth rate, you add one to it and you multiply that by your principle amount and then to do it for two years, just multiply by 1.46875 again and then do it again and then for eight years, you would raise that to the eighth power. Well, that number turns out to be roughly $ 6.93. That’s way more than $ 1.50, it should be $ 1.52, sorry. It’s way too high, it’s way too high. Okay, so that’s incorrect.
That’s not the correct rate of growth.
(Figure 2)
Figure 2, Segment 3
We could do something like this. It has risen by a total of a hundred, I don’t know why I keep trying to talk in cents, $ 1.20 in stock. So you could take that, divide by eight and add that to the price, imagine it grew at a constant rate okay, and so it would be 47 cents after the first year. The rate of change then would be this 0.46875 that we calculated earlier and then if you raise it by another $ 0.15, you get the rate of growth, it gets a little bit smaller and you can just keep working this problem, add 15 to this, add 15 again, 15 again, 15 again and you get these rates of growth. And you can see they’re going down each year. And then we could say well alright, I’ve got eight rates of growth, average them up, add them up, divide by eight. That gives me a little over 22% growth. Is that correct?
(Figure 3)
Well, let’s test that again. If I take that 22.004, add one and raise that to the eighth
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Figure 3, Segment 3 power, multiply by my principle, I get $ 1.57, that’s closer, but it’s still too big.
(Figure 4)
And I’ll be honest with you, the only reason that worked out close at all is because we did things in a nice uniform pattern. Many stocks and bonds typically grow irregularly. You know, they’ll grow big and then fall and lose in a year. So if you average those kinds of growth rates up, it wouldn’t be anywhere near the correct growth rate. What we’re doing is we’re doing it the wrong way – we’re using the arithmetic mean and that’s where you add things up and then you divide by the number of things.
That’s how you average your grades for example. So, the arithmetic mean of four, six, and eleven, you take those numbers, you add them up, and then you divided by three. The sum is 21, divide by three, you get seven. Now what that seven tells me is, that if I were to add these three numbers up I would get 21. If I replace each of these numbers with a seven,
I would still get 21. What we need actually, you’ll notice that we’re multiplying a number eight times and so we need a number that we can replace those numbers with and get the same product each time.
(Figure 5)
So we’re going to use, what we need to use is called the geometric mean and that’s where
I multiply things together, and then I take the n th root of however many numbers I have and if you prefer, you can use this kind of exponent instead of this. This is kind of hard to find on some calculators, but you can always do this. So the geometric mean of four, six, and eleven is 6.41507. That’s less than seven, that was the arithmetic mean. It’s always going to be a bit less than the arithmetic mean, but if I multiply those numbers together, I get the same product each time.
(Figure 6)
So, what we were doing is we were calculating percent change like this, we were actually
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Figure 4, Segment 3
Figure 5, Segment 3
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Figure 6, Segment 3 taking one plus the percent change.
(Figure 6) Well if I move this one over, then that’s new over old. And so I’m going to be a little vague here and just say well let a n be the price share at the end of n th year, and I’m not going to make up numbers this time. Each time this a n over the last number will give me the multiplier for that year. So when I find the geometric mean of all those things, you’ll notice how these lay out, I can cancel things and just take the last value over the first value and take the n th root, that gives me that multiplier and you’ll notice when I plug that in I’ll get $ 1.52, which is what I wanted and that means that I had a 21.5% roughly average rate of growth. Okay, and that’s an accurate measure.
(Figure 7)
We can use this application in some algebra problems. You know, if I look at this business of new over old, then I multiply both by the number of shares, well then that’s multiplying by one, so it’s not going to change value. So instead of doing the stock prices under that radical,
I could do the values of the stocks under that radical. Let me show you how that would look.
Let me show you a quick little problem like that. Let’s say you’ve forgotten how many shares you have, but you know in the last eight years that they’ve grown at an average rate of, this time we know that the average rate is 21.5%, they’re now valued at $ 1.52 cents each, and your financial guy tells you that or gal tells you that they’re actually worth $ 10,200 more now than then. So how many shares do you own?
(Figure 8)
Is that something you’ve got to go look at? You probably would have it in a lock box somewhere. Well, if you let “x” be the number of shares, then this this business here of 1.52
times “x”, that gives you then, your total value right now. And the value then is 10,200 less than that. Again, arrange those under the radical, eighth root, 100% of what you have plus this 21.5, or convert it to a decimal.
(Figure 8) You can plug those things in. That’s a hard
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Figure 7, Segment 3
Figure 8, Segment 3
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thing to solve, we won’t talk about how to solve it symbolically, but you can just flash them into the calculator and graph them.
(Figure 9) I’ll actually just zoom in on them a little bit.
Use the calculator functions to see where they intersect and sure enough they intersect right at 8,500, so you have 8,500 shares. Nice little algebra application of radical expressions.
(Figure 10)
Figure 9, Segment 3
I went through a lot of stuff really fast there so as I always do, I’ll kind of end here with some summary pages that kind of hit the key points of what I’ve talked about.
Closing
Well, the show’s over. I hope you learned some stuff, I hope you saw how we can use some of that math we’re learning for good applications. I hit things really fast and I know that. I’m squeezing a lot of content in here, but if you want to go back and view these things again, you can watch the rerun or you can go to our web page, which I’ve actually pulled up on my computer screen. It’s at Math Matters, and we’ll flash that up here at the end again, just the regular address then “/mathmatters”. On this page, you’ll find a place where you can download the episodes for one thing right here, or you can e-mail me, which I would love you to e-mail me and give me some suggestions for other programs. With that, I’m done.
Thank you so much for your time, and I’ll see you next time.
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