Expected value refresher

The expected value of an event is its most likely outcome. Assign each potential result a probability. The expected value is sum of all the potential results x their respective probabilities:

∑ (potential_result₁ x probability₁,… potential_result_n x probability_n)

Consider the simplest example possible, the coin flip. You’ll be paid R10 if you pick tails, but you’ll have to pay out R10 if you pick heads. The expected value on this one turns out to be zero – which you obviously could never get on a single flip, but if you play enough times you’re likely to “even out” in this way:

Game – Coin flip
Result	Probability	Payout	Weighted Values
Tails	0.500	R10	R5
Heads	0.500	-R10	-R5
Expected value			R0

 

Let’s look at another stupid example! You’ll be paid R10 if you pick tails, and you’ll also be paid R10 if you pick heads – a no-brainer! Whatever you do you’re going to win R10 – hooray!

Game – Coin flip, can’t lose
Result	Probability	Payout	Weighted Values
Tails	1/2	R10	R5
Heads	1/2	R10	R5
Expected value			R10

 

Now let’s graduate (naturally!) to dice games. For R7.50 a turn, you can roll a die and whatever number you throw you get double that amount in return. So if you throw a 6 you’ll get R12. By calculating the expected value we see that this is in fact a bad bet: the expected return of R7.00 is actually lower than the cost of playing at R7.50.

Game – Dice
Result	Probability	Payout	Weighted Values
1	1/6	R2	R0.33
2	1/6	R4	R0.67
3	1/6	R6	R1.00
4	1/6	R8	R1.33
5	1/6	R10	R1.67
6	1/6	R12	R2.00
Expected value			R7.00

 

What about when the probabilities vary a bit? Let’s have a look at 2 chess games, in both cases we’re going to say if I win I get R50, but if I lose I have to pay out R30. If we’re pretty evenly matched as opponents then we can say that the expected value for me from a game like this is R10.

Game – Chess (evenly matched)
Result	Probability	Payout	Weighted Values
Win	0.50	R50	R25
Lose	0.50	-R30	-R15
Expected value			R10

 

But what if I’m an ace chess player, and you’re a rank beginner – this changes the probability situation considerably, in fact I’m so confident that my expected value now increases to R50!

Game – Chess (unevenly matched)
Result	Probability	Payout	Weighted Values
Win	1.00	R50	R50
Lose	0.00	-R30	R0
Expected value			R50

 

Now let’s look at my woeful gym attendance – unfortunately no-one is paying me to go so motivation levels have been a little low lately… in fact I’m 70% likely not to go at all! But in the example below, I’m not hopeless, I may well go just over half a time per week, again, not technically possible but on average it will work out at about that much.

Game – Gym attendances per week
Result	Probability	Payout	Weighted Values
0	0.700	n/a!	0.000
1	0.150	n/a!	0.150
2	0.067	n/a!	0.134
3	0.050	n/a!	0.150
4	0.033	n/a!	0.133
Expected value			0.567

 

What about playing the Lotto? We can get odds figures from the FAQ page and use these to calculate expected value for any given prize structure… It’s been a while since I bought a lotto ticket so I can’t remember what they cost, but I’m strongly suspecting the odds are ever not in my favour :).

Game – Lotto
Result	Probability	Payout	Weighted Values
Nothing	0.97394553	–	–
2 and bonus	0.01041667	15.00	0.16
3	0.01388889	25.00	0.35
3 and bonus	0.00097276	102.00	0.10
4	0.00072939	137.00	0.10
4 and bonus	0.00003316	1 508.00	0.05
5	0.00001326	3 016.00	0.04
5 and bonus	0.00000029	203 585.00	0.06
All 6	0.00000005	8 034 918.00	0.39
Expected value			1.25