Why is it impossible to hold these probability beliefs? Why is it impossible to hold these probability beliefs?
\begin{align*}
P(a) & = 0.3 & P(a \land b) & = 0 \\
P(b) & = 0.4 & P(a \vee b) & = 0.8
\end{align*}
I know that you cant have $P(a)+P(b)$ be less than $P(a \vee b)$. But is that the only reason why these are impossible?
 A: The short answer is that it's against the rules of probability 
$$P(a) + P(b) = P(a\vee b) + P(a\wedge b)$$
It's a theorem end of story.
That's maths, in real life it's more a question of what probability means and why you should believe the rules, which is philosophically quite deep.
There's an easy answer though, if you believe the above then I can con you out of money.
Suppose you believe the above now I offer you 31p on the condition that you give me £1 if $a$ happens.  As the probability of $a$ is $0.3$ you happily accept my offer looking forward to a 1p profit.
Next I offer you 41p on the condition that you pay me £1 if $b$ happens.  The probability's $0.4$ so again you make a profit of 1p.
Finally I say I'll pay you a pound if either $a$ or $b$ happens, not for free of course, I want 79p from you now.  $P(a\vee b) = 0.8$ another 1p profit for you.
So, you've gone away thinking you've made 3p expected profit. In fact, I've paid you 72p and claimed 79p but whatever happens I can't loose money, if $a$ or $b$ happen I have to pay you £1, but you have to pay me at least £1 (maybe £2).
Now as you think you've got a good deal I can offer you the same thing again, taking 7p off you each time, I can't loose but if we keep it up for long enough I'll take all your money and you'll still think you've got a good deal.
It turns out all the basic rules of probability have equivalent "Dutch books" like the one above. So in a sense you have to believe them because if you don't you can end up acting irrationally.
A: Answer 1: What other reason do you need?
Answer 2: What happens when you test it several times? Statistically, a should happen 3 out of 10 times and b should happen 4 out of 10 times. Now think about a or b.
