Question

Discuss the law of averages and how it can be used to analyze a gambler's long run success or failure. Does the law of averages mean there's no such thing as luck?

Answer 1

Law of averages is just an informal way to define weak law of large numbers.

So what is weak law of large numbers ?

Below is one easy to understand definition.

In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value and will tend to become closer to the expected value as more trials are performed..

what does it means ?

See we know theoretically probability of observing a head or tail in a single toss of unbiased coin is 0.5.

So what happens if you toss the coin twice ? As both first toss and second toss are independent and probability is 0.5 we expect 1 head and 1 tail. So does that really happen? No.

But Law of averages tells us that if we repeat this experiment a large number of times our proportion of heads observed will get closer to 0.5 which is expected.

Now in context of gambler long run success it really depends on in which type of gamble he or she is.

lets say if we get a head we get $2 and if we get a tail we pay $1 .

So if you play this for 1 or 2 times you might end up losing some money . But if you play this for large number of times you will surely end up earning not loosing because in the long run both number of heads and tails hsould be almost equal and you are getting $2 of heads and you will be paying $1 for tails

so expected earning = $2*0.5-$1*0.5=1-0.5= $0.5

so in the long run will you will gain. So success failure depends on the type of gamble.

Law of averages doesn't mean that there is no luck . As I said it only tells us that what will happen over long run but let's say you get 3 consecutive heads so you will get $6 although the probability  of happening such events is low but not impossible you can get 3 heads in a row.

Also law of averages doesn't say anything about order you can get heads or tails in any random order so if you and your friend is play this gamble it's completely possible that one of you get's heads first and other ends up with tail so it's a matter of luck.

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