The casino La Bella Fortuna is for sale and you think you might want to buy it,

Question

The casino La Bella Fortuna is for sale and you think you might want to buy it, but you want
to know how much money you are going to make. All the present owner can tell you is that
the roulette game Red or Black is played about 1000 times a night, 365 days a year. Each
time it is played you have probability 19/37 of winning the player's bet of $10 and probability
18/37 of having to pay the player $10.
Explain in detail why the law of large numbers can be used to determine the income of the
casino, and determine how much it is.

Explanation / Answer

Answer to the question)

The law of large numbers is pretty simple.

when we experiment a trial for small number of times. we may get any probability value

but when the same experiement is repaeated a LARGE number of times, the probability values equate to the theoretical values that we already have.

For example : Suppose the experiment is to toss a coin.

the experiment is repeated 10 times

we get head 7 times , thus P(H) = 7/10 = 0.7

Though the theoretical value of P(H) = 0.5

.

Now suppose the same experiment is repeated say 1000 times, we get 570 heads

P(H) = 570/1000 = 0.570

we can observe that this value of 0.570 is very close to the theoretical value 0.50

.

Similarly if the experiment is repeated a more number of times, we will observe that the experimental probability will equate to the theoretical probability value ultimately. That is the reason why the law of large numbers helps us here

we can see that the total number of times the game is played in an year = 1000*365 = 365000

The chances of winning 19/37 and losing 18/37

Thus expected Profit = 365000 *10 * (19/37 -18/37)

Expected Profit = 3650000

This is the expected profit in case if the game is played 365000 times, and th experimental proabilities equate to the theoretical probabilities

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