How to compare observed vs. expected events?

Question

Suppose I have one sample of frequencies of 4 possible events:

Event1 - 5
E2 - 1
E3 - 0
E4 - 12

and I have the expected probabilities of my events to occur:

p1 - 0.2
p2 - 0.1
p3 - 0.1
p4 - 0.6

With the sum of the observed frequencies of my four events (18) I can calculate the expected frequencies of the events right?

expectedE1 - 18 * 0.2 = 3.6
expectedE2 - 18 * 0.1 = 1.8
expectedE1 - 18 * 0.1 = 1.8
expectedE1 - 18 * 0.6 = 10.8

How can I compare observed values vs expected values? to test if my calculated probabilities are good predictors?

I thought of a chi-squared test, but the result change with the sample size (n=18), I mean, if I multiply observed values by 1342 and use the same method the result is different. Maybe a wilcox paired test works, but what do you suggest?

If can suggest in R, it would be better.

Answer 1

You mention that you get different results if you multiply all values by $1342$. This is not a problem. You should get very different results. If you flip a coin and it comes up heads, this doesn't say very much. If you flip a coin $1342$ times and you get heads every time, you have much more information suggesting that the coin is not fair.

Usually you want to use alternatives to a $\chi^2$ test when the expected number of occurrences is so low (say, under $5$) in a large percentage of your categories (say, at least $20\%$). One possibility is Fisher's exact test, which is implemented in R. You can view the $\chi^2$ test as an approximation to Fisher's exact test, and the approximation is only good when more of the expected counts are large.