“Yes I need Methods to get into uni, but it’s so boring. When are we going to use this after graduation?”
This might surprise you, but the answer might be pretty often! Ignoring preparation for statistics, engineering and all that far-distant uni content, the things you learn in Maths Methods pop up a lot.
Australia is famous for its banks. They withstood the huge economic crash back when you were a kid, and they recently got in pretty big trouble with the government. But how do you pick which one to save your money with?
The answer is interest rates.
They’re pretty low right now, so banks need extra incentives to get you to give them your money. Here are two interest rates from different banks you’ll get as a student saver (I put them under pseudonyms, but you can find this out online):
A: 0.80% p.a. for 3 months, 0.05% afterwards
B: 1.05% p.a. for 5 months, 0.05% afterwards
From a glance at this, it looks like bank B is your best bet. But what happens if bank A offers you $10 for opening an account right off the bat, as compensation for their low-tier reward? It all depends on how much you invest.
The formula we use for calculating compound interest is all based off the exponential function, n^{x}. There’s a little section on this in your textbook, but it looks like this:
Where P is your balance, P_{0} is the principal amount (the bit you put in at the start), r is your rate per year, n is how many times a year the rate is compounded (we’re going to use n=1 for simplicity), and t is how many years have passed.
So let’s assume a family member is super rich and gives you $10,000 to put in your account, to be saved for ten years (remember, P_{0} for bank A is $10,010). Here’s how much you get out for bank A vs bank B:
Your rich family member is pretty sure the ten bucks isn’t worth it.
But I’m going to hedge my bets and say your family member, no matter how rich they are, won’t trust you with ten grand instead of putting it in a term deposit. Instead, let’s go with the assumption that you have about $500 to your name. The figures start to look a bit different:
Hey, maybe bank A’s interest rates suck, but ten bucks is ten bucks. You, an average-Joe undergrad student, will take your chances.
There’s a reason why banks do this. Sure, it looks like they’re giving money away for free, but when bank A has 14 million customers opening accounts rather heading to bank B, it’s definitely making enough off your meagre $500 to warrant it.
If you’ve ever, out of interest, looked up “how quickly can my car stop”, you might get a figure like “a decrease in velocity by 15 fps”. As frustratingly awful a figure as this is, we can convert this to nice metric units and round up:
If you’ve taken physics before, this equation for the deceleration of a car looks pretty familiar. For those of you that haven’t, there’s a chapter in your textbook about this; acceleration is the derivative of velocity with respect to time. Or in other words, we can integrate acceleration to get velocity:
Where t is how many seconds have passed, and c represents your speed just before decelerating. Let’s pick two different values for c on the fastest freeway in Victoria, the Hume Freeway: one over the speed limit (144 km/h = 40 m/s) and the one just under the speed limit (108 km/h = 30 m/s). Because we are looking at the relative distance covered, we can integrate our velocity again to produce equations for both of these without a constant:
Now let’s kick things up a notch. Let’s say you need to stop suddenly. Going 108 km/h, this would take you:
Going 144 km/h, this would take you:
And how far do you travel in each instance?
So the Transport Accident Commission calling you a “bloody idiot” is very much applicable when you realise there’s a broken-down car 100m ahead of you and you crash into it at 88 kilometres an hour.
All right, here’s something a little more fun.
If you don’t already know how to play poker, the instructions to Texas Hold’em can be found here, but you don’t need to know them very well to follow this.
Predicting how good your opponents’ hands are is difficult, so for the sake of this demonstration, I’m going to pretend that all players in a game of poker stay in 50% of the time when a raise of $100 is made, and 100% of the time when a raise of $10 is made. And to demonstrate why this matters in game theory, let’s invent two scenarios:
Scenario 1: Three players, including you.
Scenario 2: Nine players, including you.
In both scenarios, you’ve got $100 on you. If you decide to bluff, you go all-in. If not, you put in $10 and keep the remaining $90. If you win, you get back your bet multiplied by the number of people who stayed in (including you).
And let’s further suggest that you are dealt one of two similar, usually winning hands: the best full house (3 Aces, 2 Kings), where the chance p any given player draws a better hand than you is 0.0256%; and the best three of a kind (3 Aces, King, Queen), chances 0.76%.
In Scenario 1, on average, the number of people who stay in when you raise $100 is 2, and 3 when you don’t – your chances of winning are (1-p)^{1} when you do, (1-p)^{2} otherwise. Your total expected money at the end of a round looks like this:
Full House | Three of a kind | |
$100 | $199.95 | $198.48 |
$10 | $119.98 | $119.55 |
Obviously, it’s in your interests to make a large raise, because you have a winning hand. But even though the probability of getting a full house is 30 times smaller than getting three of a kind, your earnings are not that different.
However, things look different in Scenario 2. The number of people who stay in when you raise $100 is 5, and 9 when you don’t – your chances of winning are (1-p)^{4} when you do, (1-p)^{8} otherwise. Your money at the end of a round looks like this:
Full House | Three of a kind | |
$100 | $499.49 | $484.97 |
$10 | $179.82 | $174.67 |
Note that your winnings are significantly higher with more people, as is the difference between the two hands’ returns ($14.52). This is why poker has always been idealised with a high number of players and high stakes, in order to feel like your time playing is “worth it”.
Maybe you don’t have a bank account, own a car, or play poker. But as they say, three things are certain in life: tax, death, and mathematics.
hsc, mathematics, Maths, maths methods, qce, study, vce
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