Why would we look at a coin toss?

We look at coin tosses because when we toss a coin, we don't know what's coming next: heads or tails - it's a random event.

A fair coin, tossed fairly, will land 'heads' about 50% of the time and 'tails' about 50% of the time, but nobody knows which way it will land next time. Obviously it can't be heads half the time on a single toss; on a single toss, it has to be 100% heads or 100% tails. If we toss a fair coin many times, it does come up heads about half the time. Let's take a look at some aspects of that.

Suppose we toss a coin and it comes up heads. Now if we want to toss a second coin also, the second coin could come up either heads or tails. If the first coin had come up tails, the second coin could still come up either heads or tails. So there are four combinations or possibilities:

	First Coin	Second Coin
1	Heads	Heads
2	Heads	Tails
3	Tails	Heads
4	Tails	Tails
	Click to Flip!	Click to Flip!

Got that? One coin has two possibilities; two coins have four possibilities.

Ready to go on?

So how many ways could three coins come up?

Suppose the third coin came up heads. The other two coins could come up the ways described before (4 ways). If the third coin came up tails, the other two coins could again come up the ways previously descibed (the 4 ways again), totaling eight ways.

	Third Coin	First Coin	Second Coin
1	Heads	Heads	Heads
2	Heads	Heads	Tails
3	Heads	Tails	Heads
4	Heads	Tails	Tails
5	Tails	Heads	Heads
6	Tails	Heads	Tails
7	Tails	Tails	Heads
8	Tails	Tails	Tails
	Click to Flip	Click to Flip	Click to Flip

So what is the rule? Each coin has two possible states, and so every additional coin multipies the number of possibities by 2 (one for heads; one for tails).

Combinations = 2^{Number of coins}. Remember from the Exponents lesson, that the exponent (Number of coins) indicates how many times the base (2) is multiplied.

Why 2? Because each coin has two possible states. And this is important because some random elements have more than two states. We'll learn about that in Lesson 3.

A standard die (singular of dice) is a cube and so has six sides. If it is a fair die, each side is equally likely to come up on a toss. This means there is a one-in-six chance that any particular number will come up on the next toss.

Imagine we have two dice now. If the first came up a '1', the second could come up in any of six ways. We see that with two dice, the number of combinations is 6 times 6, or 36.

And the same rule applies: every additional die multipies the number of possibities by 6.

Combinations = 6^{Number of dice}.

If we had a ten-sided die, say, numbered from 0-9, there would be a one-in-ten chance of the tossed die coming up with any particular number.

Let's do an experiment

How many times would you have to roll a die to get a particular number? If there are 10 possible ways the die could come up, then over many rolls of the die, each value should come up 10% of the time. For each of the below, click until the specified number comes up.

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Click until it comes up 0,

0
Click to try to roll a 0

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Click until it comes up 1,

1
Click to try to roll a 1

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Click until it comes up 2,

2
Click to try to roll a 2

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Click until it comes up 3,

3
Click to try to roll a 3

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Click until it comes up 4,

4
Click to try to roll a 4

If we repeated this experiment many, many times, and took the average of all those averages, it would be very close to ten. Your results on one test might be far from ten, but another sampling might be very different. However, the average of a lot of samples will be around ten. Why? Because there are ten sides to that die.

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If we tossed two such dice, there would be 10² possible ways the pair could come up.

Let's do another experiment

Let's see what random really looks like. You know the number π - it's a non-repeating, non-terminating number bigger than 3 but smaller than 4.

 to try to randomly select the first few digits of π.

3	.	1	4	1	5	9	2	6	5	3	5...

Is that a good approximation of π? It's rare when a random event produces what you want, and the more precise the thing you want, the more rare it is.

And consider this: another variable we didn't take into account is the location of the decimal point!

Knowledge Check

Imagine you have 1 coin and 1 ten-sided die. How many combinations are there between a coin and a ten-sided die?

Think of it this way: there are 10 ways the die can come up if the coin is heads,
and there are 10 ways the die can come up if the coin is tails.

To get the total possibilities, the number of possibilities of one random element (two for the coin) is multiplied by the number of possibilities on the next random element (ten for the die).