Expectation

In Probability, picking a card from a pack to find out the outcome, or tossing a coin to see if it lands on heads or tails or any similar action is called an experiment.

The result of the experiment is called the outcome.

The same experiment, like rolling a dice, can be performed many \times. Each time it is performed is called a trial.

So you can perform 100 trials of an experiment where you toss a coin in the air to see if the outcome is heads or tails.

Frequency is the number of \times something happens.

When you know the probability of an outcome, you can predict how many \times you would expect that outcome to occur in a certain number of trials. This is called the expected frequency.

An example of this is tossing a coin 50 \times. In these 50 trials how many heads would you expect? The probability of a head is 0.5. The expected frequency of heads is 0.5 x 50 = 25.

Another example is rolling two dice and adding the numbers they land on together, The sample space of the result looks like this.

The outcome from a trial of rolling the two dice can be 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 or 12.

If X is the possible result of a trial then X = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

P(X = 2) = 1/32, P(X=3) =2/32 and so on.

If x = the actual result of a trial then

This is just another way of displaying the sample space information. It is saying that in the 36 squares of the sample space, one is a 2, two are a 3, etc.

This information can be written like this

And can be plotted like this

The plot shows clearly that 7 is the most likely outcome of a trial. It also shows that the probability distribution is symmetrical.

If you rolled two dice 144 \times, how many \times would you expect the sum of the two dice to add up to each of the possible outcomes?

We would expect the dice to give a sum of 2 once every 36 rolls. In 144 rolls we would expect 144 x 1/36 = 4

We would expect a sum of 7 once every six rolls. 144 x 1/6 = 24

In a similar way, we can calculate the expected frequencies for all of the possible outcomes

This can be plotted too.

Notice that the shapes of the probability plot and the expected frequency plot are the same (only the y-axes are different). This makes sense. The greater the probability of an outcome then the greater the expected frequency from a certain number of trials. The expected frequency is directly related to the probability and so the plots have the same shape.