Intresting Probability problem Suppose that, over the course of the next year, a particular investment fund has a 40% probability of beating the market and a 60% probability of under-performing (perhaps they are poorly managed, or charge high fees). If the fund outperforms the market it will (certainly) continue operating for another year, but it is in danger if it under-performs: in that case there's a 50% probability that its investors will angrily withdraw their money, so that the fund simply ceases to exist. If the fund still exists at the end of the year, what's the probability that it beat the market? You'll find that it's higher than the prior probability of 40%.

Ah, this is a problem regarding conditional probability. We define a probability space $\Omega = \\{a, b, c\\}$, where

* $a$ means that the fund has beat the market,
* $b$ means that the fund has not beat the market, and the investors did not withdraw,
* and $c$ means that the fund has not beat the market, and the investors withdrew their money.



Now from the assumption, it follows that if $P$ is the probability measure, then $P(\\{a\\}) = 0.4$, and $P(X) = 0.6$, where $X = \\{b,c\\}$ is the event that the fund underperformed. Moreover, $$ P(\\{c\\}|X) = 0.5 = P(\\{b\\}|X). $$ But since $\\{c\\} \cap X = \\{c\\}$ and $\\{b\\} \cap X = \\{b\\}$, we may deduce from that that $$ P(\\{c\\}) = P(\\{b\\}) = 0.5*0.6 = 0.3. $$

Now note that the event that the fund still exists is $Y := \\{a, b\\}$. Then $P(Y) = 0.4 + 0.3 = 0.7$ and moreover

$$

P({a}|Y) = P({a}|Y)/P(Y)=0.4/0.7 = 4/7 > 1/2,

$$

which is what we were looking for.