Bang or No Bang Game 1: Heads or Tails

Go and find a coin. Any old coin will do. You can see that any coin has two sides, heads and tails. This makes the throw of a coin a good way to decide on something, if you can't make your mind up. Have you ever said to someone; "I don't know if I want chocolate or vanilla ice cream - let's toss a coin to decide"?

Whenever you spin a coin, it is random whether it falls heads or tails. Of course you could make it fall as heads or tails (by throwing it in just the right way) but that's cheating.

You can calculate the probability of a coin landing on heads, say, by dividing the number of ways that it can fall heads (1) by the total number of ways that it can fall (2). The probability of landing on heads is therefore 1 divided by 2, or 1/2. Sometimes people say that there is a "one in two chance" of landing on heads.

More generally, the probability of something happening is:

(the number of favourable outcomes) divided by (the number of all possible outcomes)

The probability of a coin landing on heads is exactly the same as the probability if it landing on tails. If you don't believe this, throw your coin and try to guess how it will fall. No matter what you guess, you'll be right half the time and wrong half the time.

In fact, this isn't quite true. You could toss the coin ten times and it might land on heads six times, and on tails only four. So does that mean your coin isn't fair? No, it just the law of average at work. The law of average says that, even if it's a one-in-two chance of throwing heads, it doesn't necessarily mean that exactly half of the times you throw a coin that it will be heads. It just means that, on average, this is true.

Don't be too surprised when you toss a coin three times and it comes down heads each time. This imbalance will average out if you throw the coin a few more times.

So what is the probability of throwing two heads in a row? Or ten?

Let's say you decide to throw a coint ten times. (Incidentally, it doesn't matter if you do this one after the other with the same coin, or all at once with ten different coins). What kind of result could you get? Well, you might throw HTHTTHHHTH. Or TTTTHHHHTH. Or any one of a large number of possible outcomes. How many different answers could you get?

With one coin, there are two possible answers (H or T). If you throw a coin twice there are four possible answers (HH, HT, TH or TT). By the way, it's important that you note the order of the results: HT is a different result from TH... Throwing a coin three times there are eight possible results (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT). Write down all the possible results for four throws(there are 16 possible combinations) and five throws (there are 32 possible combinations).

If you throw a coin n times, there are 2ⁿ possible combinations. So for ten throws there are 2¹⁰ different combinations. 2¹⁰ = 1024. So if you toss a coin ten times there are 1024 different results you could get.

So what is the probability of throwing ten heads in a row (HHHHHHHHHH)? There is only one way this can happen (one favourable outcome) and 1024 possible outcomes, so the probability is 1/1024 or less than one-in-a-thousand.

This can be written as P(ten heads) = 1/1024 = 0.00098

So what is the probability of getting exactly eight heads when you throw a coin ten times (or when you throw ten coins)?

To work this out you need to know the number of ways that you could get exactly eight heads (like HHHHHHHHTT or THHHHHHHHT for example). To work this out you need to know about the mathematical function called "factorial". The factorial function is writen as !

3! = 3 x 2 x 1 = 6

4! = 4 x 3 x 2 x 1 = 24

10! = 10 x 9 x 8 x ... x 2 x 1 = 3628800

As you can see, applying the factorial function to a number you can get very bigs answers (which is maybe why it is written "!")

The number of ways of throwing exactly eight heads in ten coins is:

10! / (8! x 2!)  = 45

In general, the number of ways for choosing r heads in K tosses is

K! / ((K-r)! x r!)

This equation lets you know the number of favourable outcomes, which allows you to work out the probability.

The probability of throwing exactly r heads in K tosses is

(the number of favourable outcomes) divided by (the number of possible outcomes)

or

(K! / ((K-r)! x r!)) divided by (2^K)

Work some out for yourselves!

(a) what is the probability of throwing exactly six heads if you throw a coin ten times?

(b) what is the probability of throwing exactly twelve heads if you throw a coin twenty times?