Step by step
The question
Question: x can be 1, 2, 3, 4, with probabilities 0.1, 0.3, 0.4, 0.2: find the mean.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Check the table
x is a random variable: a number that comes out by chance.
The table lists each value of x and its probability, P of x.
A probability table must add up to 1.
0.1 plus 0.3 plus 0.4 plus 0.2 is 1.
Good.
2 Multiply x × P(x)
The mean of a random variable: each x times its probability, all added up.
Why: in many tries, each x comes up a P of x share of the time.
So that is its share of the average.
1 times 0.1 is 0.1.
2 times 0.3 is 0.6.
3 times 0.4 is 1.2.
4 times 0.2 is 0.8.
3 Add the column
0.1 plus 0.6 plus 1.2 plus 0.8 is 2.7.
The mean, μ, is 2.7.
Careful!
The plain average treats every x the same.
But the likely ones count more: we weight by P of x.
4 Check
Let's check with 10 tries.
Each x comes up about 10 times its probability: 1, 1 times; 2, 3 times; 3, 4 times; 4, 2 times.
All the x's of the 10 tries add up to 27.
27 divided by 10 is 2.7: the mean.
So it is right!
5 Pick the choice
Our answer is μ = 2.7.
That is choice B.
6 Why not the others
A is 2.5.
Wrong: the mean of 1, 2, 3, 4: weight each value by its probability.
That is the Careful mistake from before.
C is 3.
Wrong: the most likely value.
D is 0.25.
Wrong: the mean of the probabilities.
Let's look back.
First: check the table.
Then: multiply x × P(x).
Then: add the column.
Then: check.
Then: pick the choice.
Last: why not the others.