Calculating the Probability that a Soccer player with a high salary scores a lot of goals

purplesoup
purplesoup used Ask the Experts™
on
Suppose I have a table of soccer player salaries (in terms of high/medium/low):

High: 99
Medium: 165
Low: 157

I also have a table of soccer player goals scored (again in terms of high/medium/low):

High: 167
Medium: 112
Low: 142

And suppose I have a third table showing a breakdown of soccer player salaries and goals:

                                                Salaries
                              Low             Medium        High
          Low              70                   52               20
Goals Medium       43                   47               22
          High             44                   66               57


Now suppose I want to calculate the probability that if I pick a soccer player at random, what is the chance that I will pick one with a high salary who scores a lot of goals?

It would seem there are two ways of calculating this - one from the separate goals and salaries tables - find the chance of picking one with a high salary, and combine it what the probability of picking one who scores a lot of goals, or look at the combined table and calculate the probability from that table.

The trouble is, each method gives a different answer.

Is there a way of finding what the correct answer is?
Comment
Watch Question

Do more with

Expert Office
EXPERT OFFICE® is a registered trademark of EXPERTS EXCHANGE®
Most Valuable Expert 2011
Top Expert 2012
Commented:
You have 57 players that match your criteria of high salary and high goals


you have 167 + 112 + 142  total players  (or add the salary counts, same result)

so...

57 / ( 167+112+142)
You might want to leave the goal keepers out of your calculations, some of those are very highly paid and are still described as soccer players.
Awarded 2010
Top Expert 2013
Commented:
The two ways will get you the same answer if you weight it properly.
For example, chance of low/low
70/(total) = 70 / 421 ~ 16.6%

Chance of low salary = 157/421
Chance of low pay given low salary = 70/157
Combine them (70/157) / (157/421) = 70/421 ~ 16.6%
Microsoft Azure 2017

Azure has a changed a lot since it was originally introduce by adding new services and features. Do you know everything you need to about Azure? This course will teach you about the Azure App Service, monitoring and application insights, DevOps, and Team Services.

purplesoupProgrammer

Author

Commented:
sdstuber - here's my problem. Yes you can just use the main table an do 57/142 = 0.13 2dp.

But using the other two tables the chance of getting a high salary is 99/421 and the chance of scoring a high number of goals is 167/421 so shouldn't combining those two figures also get me to 0.13, but it doesn't, it works out as 0.09 2 dp.

TommySzalapski - sorry I'm still not getting it. I can see that the other two smaller tables are actually just repeating the data in the main table - so if you add up all the low salaries in the main table you get 157 which is in the small table, and if you add up all the low scorers in the main table you get 142, the same as the figure in the small table.

So I can see that it is possible to combine the chance of being a low scorer or a low salary in one of the smaller tables, and combine that matching row or column in larger table, to get a consistent value between the two.

My confusion - and clearly I'm wrong here I just can't see why - is that it would have been possible to use the data from the two smaller tables to get the same value as from the larger table.

Put simply, it is possible to find out - just using the two smaller tables - the chance of picking a player with a high salary and high number of goals (or low salary and low number of goals), with a value that is consistent with calculating this figure from the larger table?
Most Valuable Expert 2011
Top Expert 2012
Commented:
The different tables don't lead to different results.

57/142   --- this is not correct


57 / ( 167+112+142)    --- this is correct


you have (167+112+142)   total players so 421.

Either your scoring table or your pay table will tell you that or the 3rd table if you count up all 9 categories.

Your 3rd table tells you 57 of them are high pay and high scoring
i.e. what you are looking for

So the chance of picking one of them is 57/421

You can't use either the first or the second without the third
Because there is nothing that connects salaries with scoring.

Can use the third by itself because it has the total count and the count of the category you need.
ozo
Most Valuable Expert 2014
Top Expert 2015
Commented:
Number of  players with a high salary who scores a lot of goals / total players  = 57/421

Number of  players with a high salary / total players = 99/421
Number of  players with a high salary who scores a lot of goals / Number of  players with a high salary = 57/99

Number of  players who scores a lot of goals / total players = 167/421
Number of  players with a high salary who scores a lot of goals /
Number of  players who scores a lot of goals = 57/167


57/421 =  (99/421) *  (57/99) = (167/421) * (57/167)
all three methods give the same answer
Most Valuable Expert 2014
Top Expert 2015
Commented:
Using only the two smaller tables of
Suppose I have a table of soccer player salaries (in terms of high/medium/low):

High: 99
Medium: 165
Low: 157

I also have a table of soccer player goals scored (again in terms of high/medium/low):

High: 167
Medium: 112
Low: 142
you won't have enough information to distinguish between

                                                Salaries
                              Low             Medium        High
          Low              70                   52               20
Goals Medium       43                   47               22
          High             44                   66               57

or
                                                Salaries
                              Low             Medium        High
        Low              60                   42               40
Goals Medium       33                   37               42
          High             64                   86               17

or
                                                Salaries
                              Low             Medium        High
        Low              80                   62                0
Goals Medium       53                   57               2
          High             24                   46               97

or
...
purplesoupProgrammer

Author

Commented:
Ok thanks - that's clear.

I had a typo in my previous reply - I wrote
"sdstuber - here's my problem. Yes you can just use the main table an do 57/142 = 0.13 2dp."

I should have written 421 instead of 142 - the answer is right (i.e. 57/421 = 0.13 2 dp).

Anyway, for the other replies, thanks for the clarification - the other two smaller tables are irrelevant to answering the question, although it is possible to use the data to confirm the answer, however all the information that is needed is in the larger tables.

I assume the reason you can't just multiply the two values from the smaller tables together is that they aren't independent?

Anyway, I'll share out the points, thanks again.
Awarded 2010
Top Expert 2013

Commented:
That is correct. If they were independent events then the expected values for the big table would be the products of the probabilities of the smaller tables.
purplesoupProgrammer

Author

Commented:
Thanks - thinking about it some more that's obvious really.

If we have two people, 50% are tall and 50% are bank managers, we can't say there is a 25%  probability that picking one at random will be a tall bank manager because we don't know if the tall person is the bank manager or not.

On the other hand if there is a 50% chance I will meet a tall person at the gym on Monday and a 50% chance I'll meet a Bank Manager on my train journey on Tuesday, there is a 25% change I'll meet both.

Thanks again.

Do more with

Expert Office
Submit tech questions to Ask the Experts™ at any time to receive solutions, advice, and new ideas from leading industry professionals.

Start 7-Day Free Trial