Arsenal Collection: Who's the Winner?

Thank you to Anna from Mt Eliza North Primary School, Australia, who sent us this solution (originally to another version of this problem):

1. What are the possible results if 2 goals are scored in total?

2-0, 0-2, 1-1.

2. Why are they not all equally likely?

They are not all equally likely because there are two ways you can get the end result, 1-1, by [Arsenal] scoring first and then [Chelsea] or [Chelsea] scoring first and then [Arsenal]. There is one way for the end result, 2-0, [Arsenal] gets the 2 goals without [Chelsea] getting 1. And there is only one way you can get the end result 0-2, by [Chelsea] getting both the goals. Therefore the end result 1-1 is more likely.

3. Is this a reasonable assumption?

Yes. Even if [Arsenal] are more confident and [Chelsea] are trying harder, those equal out so the chance would stay at 50-50.

4. What are the probabilities of each result according to Alison's model?

In Alison's model the team who scores first is twice as likely to score the next goal.

The chance of ending 2-0, with the teachers winning, is:

$\frac{1}{2}$$\times$$\frac{2}{3}$$=$$\frac{1}{3}$

so there would be a 33.3% chance of that occurring.

The chance of ending 2-0, with [Arsenal] winning, is also 33.3%:
 
$\frac{1}{2}$$\times$$\frac{2}{3}$$=$$\frac{1}{3}$ 

The chance of ending 1-1, with [Arsenal] scoring the first goal and [Chelsea] scoring the second goal, is:

$\frac{1}{2}$$\times$$\frac{1}{3}$$=$$\frac{1}{6}$

The chance of ending 1-1, with [Chelsea] scoring the first goal and [Arsenal] scoring the second goal, is also:

$\frac{1}{2}$$\times$$\frac{1}{3}$$=$$\frac{1}{6}$

So altogether the chance of ending 1-1 is also 33.3%:

$\frac{1}{6}$+$\frac{1}{6}$=$\frac{1}{3}$


Therefore, using Alison's model there is a 33.3% chance of any of the three results occurring.

5. What are the probabilities of each result according to Charlie's model?

In Charlie's model, after a team scores, the opposing team are twice as likely to score the next goal, because they start trying harder.
 
The chance of ending 2-0, with [Arsenal] winning, is:

$\frac{1}{2}$$\times$$\frac{1}{3}$$=$$\frac{1}{6}$

so there would be a 16.7% chance of that occurring. 
 
The chance of ending 2-0, with [Chelsea] winning, is also 16.7%:

$\frac{1}{2}$$\times$$\frac{1}{3}$$=$$\frac{1}{6}$  
 
The chance of ending 1-1, with [Arsenal] scoring the first goal and [Chelsea] scoring the second goal, is:

$\frac{1}{2}$$\times$$\frac{2}{3}$$=$$\frac{1}{3}$

The chance of ending 1-1, with [Chelsea] scoring the first goal and [Arsenal] scoring the second goal, is also:

$\frac{1}{2}$$\times$$\frac{2}{3}$$=$$\frac{1}{3}$

So altogether the chance of ending 1-1 is 66.6%:

$\frac{1}{3}$+$\frac{1}{3}$=$\frac{2}{3}$ 

Therefore, using Charlie's model there is a much greater chance of ending up with a 1-1 result.


Krystof from Uhelny Trh, Prague, used tree diagrams to work out the probabilities.