Pascal’s Triangle with Probability

Pascal's Triangle

Last week in this series we used a web or network of nodes and found our way to Pascal’s triangle.

Today we’re starting with the old coin-flipping experiment. You must have done one of those, where you flip coins 100 times and end up smelling of pennies? Well, we’re not going to do that! Not even with gloves on to avoid the smell.

Here’s a picture of what can happen if you flip one coin:

1 coin results

You can get a head with the first coin or a tail. (This is where the 1/2 probability of flipping a head or a tail on a coin come in – one chance out of two results, so 1/2.)

How about two coins?

2 coin results

Both heads (HH), a head and a tail (HT), a tail and a head (TH), or both tails (TT). Can you see how the middle two pictures are different? This means that you need to count them as different possibilities, there are four different results, one gives two heads, two give a head an a tail, one gives two tails. (Hence the probabilities: two heads has a probability of one out of four, 1/4; a head and a tail in any order has a probability of 2/4 = 1/2; two tails has a probability of 1/4.)

How about three?

All heads (red label), three different ways for two heads and one tail (orange writing), three different ways for one head and two tails (green label), or all tails (blue label.

Shall we write these options out in a table? I’m writing “the number of ways of getting an outcome”.

1 1
1 2 1
1 3 3 1

Looking familiar yet?

Possibilities for a set of four coins (I didn’t fancy making a picture of this one…) are:

There is 1 way of having four heads, 4 ways to have three heads and one tail, 6 ways to have two heads and two tails, 4 ways to have one head and three tails, and 1 way to have four tails. Next line of Pascal’s triangle: 1 4 6 4 1

Now 5 coins… Well, I’m not actually going to write out all of the options! If we take line 5 from Pascal’s triangle we see it says 1  5  10  10  5  1. This means that if we took 5 coins, there would be 1 way to have all 5 as heads, 5 ways to have four heads and one tail, 10 ways to have three heads and two tails, 10 ways to have two heads and three tails, 5 ways to have one head and four tails and 1 way to have all five coins turn up as tails.

So Pascal’s triangle gives us the number of ways of choosing a certain combination from a set of objects!

Why is this useful? In the coin-flipping experiment, we can find a probability of a certain result without having to lay out all the coins or write out all the options. Trust me, that is much easier! How do we actually do this? Let’s say your friend has challenged you to flip 5 £1 coins and get 3 heads. If you do, you get to keep the £5! (Are you paying attention yet?)

Now, before you get too excited, your friend has said you need to pay £2 for the chance to win his £5. Is it worth it? Well, how likely are you to win? Looking at the options we wrote out earlier, there are 10 ways to get 3 heads and 2 tails when you flip the coins. To find the total number of options, we add up the whole row of the triangle. 1 + 5 + 10 + 10 + 5 + 1 = 32. Your probability of winning your friend’s £5 is 10/32 = 5/16 = 31.25%. I wouldn’t bother if I were you, since your friend will keep his £5,  68.75% of the time.

Some more advanced probability stuff gives an expected value for your friend’s winnings. Whatever happens, he keeps your £2. Only 10/32 times will he lose his £5. Expected value per game is 2 + (10/32 x -5) = 2 – 1.5625 = 0.4375. In other words, every time he gets someone to play, he expects to be up nearly 44p. It is definitely not worth it to you! For your friend on the other hand…

So are you feeling the excitement yet? Pascal’s triangle isn’t just a pretty triangle with numbers, it is useful! We have linked networks and probability, two very different areas, with Pascal’s triangle.

Next week we will look at some of the patterns in Pascal’s triangle. Bonus marks if you can spot 3 or more before then!


Pascal’s Triangle series:

Pascal’s Triangle with Routes

Pascal’s Triangle with Probability

Pascal’s Triangle Sequences part 1

Pascal’s Triangle Sequences Part 2

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