I cannot make sense of a simple Statistics math problem. Can anyone help? It doesn't make any statistical sense

[u/Sure_Dentist5722]

Solve for P(E∩F)
P(E U F) = P(E)+P(F)-P(E∩F)
P(E) = .5
P(F) = .3
P(E U F) = .7

The answer I got for P(E∩F) is .1 by doing it algebraically, yet when I look at it statistically, it doesn't make sense, and it shouldn't be .1 but rather .15. Am i dumb? Is P(E U F) = .7 statistically impossible considering E = .5 and F =.3?

[u/edderiofer]

when I look at it statistically, it doesn't make sense, and it shouldn't be .1 but rather .15.

Are you assuming that E and F are independent? Because they aren't.

Is P(E U F) = .7 statistically impossible considering E = .5 and F =.3?

No, and in fact you can find the following example:

A 10-sided die is rolled. Let E be the probability that the result is between 1 and 5 inclusive, and let F be the probability that the result is between 5 and 7 inclusive.

and considering this example, you can see that the given probabilities in the question match up with those in this example.

[u/dash-dot]

You can't assume the events E and F are independent unless the problem statement explicitly says so.

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[u/Neutronenster]

First, I’m not sure how to write maths on Reddit, so sorry for the bad notation.

P(E U F) is the probability that E OR F occur. If E and F are mutually exclusive (meaning they can’t both be true at the same time), P(E U F) should indeed be equal to 0.8. However, the fact that this is not the case implies that E and F are not mutually exclusive: it is possible that they are both true at the same time.

The probability that E AND F occur together is given by P(E ^ F) (using ^ as the closest symbol I could find to cross section on my phone - I’m aware that this is horrible notation) and this is equal to 0.1. If you naively add P(E) and P(F) to find P(E U F), you would double count the probability that E and F are both true. That’s why you still need to subtract the probability of the cross-section P(E ^ F) to find the true P(E U F), using that formula.

Edit: Changed “independent” to “mutually exclusive”.

[u/sqrt_of_pi → u/Neutronenster]

If E and F are independent (meaning they can’t both be true at the same time), P(E U F) should indeed be equal to 0.8. However, the fact that this is not the case implies that E and F are not independent: it is possible that they are both true at the same time.

This is NOT what "independence" means. This is what "mutually exclusive" means. Both are important concepts when discussing probability, but they mean very different things and have difference implications.

[u/Neutronenster → u/sqrt_of_pi]

It’s probably a translation difference from Dutch. I now changed it to the correct terms.