Probabilities are written as numbers between zero and one. A probability of one means that the event is certain. If you toss a coin, it will come up a head or a tail. So there is a probability of one that either of these will happen. A probability of zero means that an event is impossible. If you toss a coin, you cannot get both a head and a tail at the same time, so this has zero probability. Anything that can happen but is not certain is written as a number less than one. It could be a decimal, a fraction, a percentage, or described as "one in a thousand", which is another way of writing a fraction. All these are ways of describing probabilities. The coins experiment and dice experiment pages use percentages. Much of the theory is easier in fractions.
There are a couple of important points. Firstly, probabilities do not tell you what is going to happen, they merely tell you what is likely to happen. It is unlikely that you will toss twenty coins and that they will all come up heads. But if enough people toss enough coins for long enough, then this may well happen. It will startle the person it happens to, but think of all the people it didn't happen to! Secondly, if you toss a coin nineteen times and it comes up heads each time, then it is not more likely that the next toss will be a tail. The odds stay the same, at 50%. The tosses are called 'independent events' which means that the coin can't remember what has happened to it. While twenty heads in a row is unlikely, once you have nineteen heads in a row, the unlikely event has already happened. The potential twentieth head has the same probability as the first head. Another way of looking at it is that any sequence of twenty tosses is unlikely as twenty heads in a row, even if it looks random. But you have to write down the sequence before you start tossing to see if you get it!
What are the possible outcomes?
When we toss a coin, there are two possibly outcomes. It can be a head or a tail, which are both equally likely. If we toss two coins, there can be two heads, two tails, or a head and a tail. It is tempting to say that there are three equally possible outcomes. But this would be wrong. You must think of the coins separately. It might be easier to imagine tossing one coin first and the other after (or even tossing the same coin twice, which has exactly the same effect). Or you could imagine two different values of coins, so they can be told apart. Now you can see that there are four possibilities: both heads, both tails, first coin a head and the second a tail, and first coin a tail and the second a head.
How many possible outcomes?
You can see that the number of possible outcomes gets bigger and bigger. Click on One more coin to see how the number of possible outcomes increases:
It is quite easy to find the different outcomes, since they are represented by the binary numbers with that amount of digits, with H representing the digit one and T representing zero. This also shows us how many outcomes there are, since there are 2n possible binary numbers with ndigits.
Working out probabilies by counting
Once you have listed all possible outcomes, then you can work out the probabilities quite easily. Say that you are going to toss three coins, and you want to work out the probability of only one head (and so two tails). The possible outcomes are:
TTT, TTH, THT, THH, HTT, HTH, HHT, HHH
All these outcomes are different, and they are all equally likely. There are8 of them. There are 3 tosses with only one head:
TTH, THT, HTT
So the probability is 3/8. You can convert this into a decimal 0.375 or a percentage 37.5%, which you can round to 38% if you wish. Or you can describe it as a three in eight chance. All these mean the same.
You can list the possible outcomes above for any dice up to 6 and count the tosses which match the probability that you want. Then divide it by the total number of throws. Or you can use this calculator, where the computer counts the number of tosses for you!
Now you can work out the probabilities for various combinations for heads and coins, then you can experiment to see if reality matches the probability! You will find it won't necessarily match (although the more numbers of throw, the closer it will get), but the experiments should give figures fairly close to the probability.
Calculating probabilities
The counting method works, and is very good for getting the right answer with a small number of coins. However, for larger numbers, we need a more mathematical approach. We know that the probability will be a fraction, and we know that the denominator (the number underneath) is2n for n coins. The problem is working out the numerator (the number on top). We don't want to count all the cases where it happens. What we can do is start with one coin, then add a coin at a time, and see what difference it makes to the probability. This starts to build up a pattern.
| The probabilities for throwing a single coin are obvious: |
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| Now we add a coin. It will either be a head or a tail. We consider the probabilities separately. If it's a head, then the result for both coins will be either 2 heads or 1 head. If it's a tail, then the result will be either 2 tails (and so no heads) or 1 tail (and so 1 head). We can write down these probabilities, and get the final probability by adding them. |
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| We add a third coin. The logic is the same, except we have more cases to consider. But you can see that we're just copying the probabilities from the previous case. |
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| We add a fourth coin. |
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| We add a fifth coin. |
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Pascal's triangle
The pattern that emerges from the above example is something called Pascal's triangle (after Blaise Pascal a French mathematician). You work this out like this. Every line is made by adding the two numbers in the line above. You assume that there are zeros at the start and end of each line, and you start with a one in the top row. Pascal's triangle crops up a lot in probability. In particular, take the nth row of Pascal's triangle, and the mth number in it, and you have the numerator (top bit) of the probability of finding m heads when tossing n coins.
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