# How to interpret the Expected Value?

Say the cost of a ticket is 1 dollar.
The odds of winning is 1 out of 228 million.
The prize is 300 million dollars.
So, the expected value is 0.32 dollars.

I'm not sure how to interpret this "expected value." What does it mean? That I can expect to make 0.32 dollars each time I buy a ticket? wut? lol

• How did you get $\$0.32$? – TheSimpliFire Feb 16 '18 at 10:34 • In this kind of situations (individual cases) it doesn't mean anything. In other contexts, especially when you are considering a large ensemble (population), like gas particles, you can use the expected value – Yuriy S Feb 16 '18 at 10:35 • Over the course of trying many tickets you will profit about 0.32 dollars each. As the number of tickets goes to infinity this becomes exact. Note that to get this level of accuracy you would need to try so many tickets that you win many different times. – Ian Feb 16 '18 at 10:35 • @Ian, I wouldn't recommend buying an infinite number of lottery tickets – Yuriy S Feb 16 '18 at 10:36 • I assume that you calculated$228/300\approx 1.32$. The expected value is actually$1.32$dollars in this case. This kind of lottery is not very common, as the expected win per ticket is larger than the price of the ticket. – Matti P. Feb 16 '18 at 10:36 ## 3 Answers If the probability of winning is$p=\frac{1}{228} \cdot 10^{-6}$and the prize is$x= \$300 \cdot 10^6$ then your expected value will be just $$x \cdot p=\frac{\ 300}{228}=\frac{\ 25}{19} \approx \ 1.316$$ Which means if you buy $228$ million tickets, you might get $\$ 1.316$for every ticket, yes. If you win. But this is just one lottery, so you can only win one single time. There's a chance you will not win and won't get any revenue for your money. But to make winning more likely, you'd have to participate in hundreds of such lotteries, and buy$228$million tickets each time. Then you will really get approximately this expected value for each dollar you've spent. If there are$228$million tickets costing$\$1$ each and exactly one of them will win $\$300$million then the ticket purchasers will collectively gain a net$\$72$ million

That is an average net gain of almost $\$0.32$per ticket The interpretation is meaningful from the point of view of the lottery company. They do have a large number of experiments and$1\$ minus the expected value times the actual number of participants is their profit with high accracy.