## Basic Probability, part 6

### August 24, 2009

The another type of presentation the probability is a table with occasions and their frequencies. For example, we have been watching basketball game and the team is stable with 5 players, and we calculated that players scores in a row are following:

If we are watching a random scene of the games and our team scored. The probability that this score has done by player 3 is

**Expectation.**

Consider an example, there is a one-armed bandit machine in a casino and it works on following idea: on every 400 attempts to win machine gives to win 2 times with $25 and 1 time with $45. If each attempt costs $1, what we can expect as average winning in long run?

To answer on this question we need to know the notion “expectation”. **Expectation or expected value **is a long run average and is calculated as:

where is the first occasion in numbers and is the probability for the first occasion, and so on for the rest occasions.

Now let us look at our example in a casino. Denote the first occasion as lose, so doll. and , the second occasion is win $25, so doll. and , the third occasion is win $45, so doll. and . Put all information together in the table:

Calculate expected value or expectation:

Round the value to

And what does it mean? It means that if we are going to play with the machine for a long time every arm dropped will cost to us $ It is the average value that we loose.

Pretend the following case.

We has sat all the night in a casino and played in one-armed bandit. We dropped the arm 400 times, on 25th and 142nd attempt we won $25, on 177th drop we won $45, the rest attempts were lose.

You can see that sometimes we lose, sometimes we won, however it looks like we came to the machine and gave $ for every attempt without losing $1 or winning $25 and $45.

Now let us see the view from the casino owners. When one is going to play in our casino in one-armed bandit and saying “I am going to win”, he(she) is wrong. We are going to win! And at an average on every drop we win $ or approximately 66 cents.