Interpreting expected value | Probability & combinatorics | Khan Academy

We're told a certain lottery ticket costs two dollars, and the back of the ticket says the overall odds of winning a prize with this ticket are 1 to 50. The expected return for this ticket is 95 cents.

Which interpretations of the expected value are correct? Choose all answers that apply. Pause this video, have a go at that right now.

All right, now let's go through each of these choices.

So choice A says the probability that one of these tickets wins a prize is 0.95 on average. Well, I see where they're getting that 0.95; they're getting it from right over here. But that's not the probability that you're winning; that's the expected return. The probability that you win is much lower. If the odds are 1 to 50, that means that the probability of winning is 1 to 51. So it's a much lower probability than this right over here. So definitely rule that out.

Someone who buys this ticket is most likely to win 95 cents. That is not necessarily the case either. We don't know what the different outcomes are for the prize. It's very likely that there's no outcome for that prize where you win exactly 95 cents. Instead, there's likely to be outcomes that are much larger than that with very low probabilities. And then when you take the weighted average of all of the outcomes, then you get an expected return of 95 cents. So it's actually maybe even impossible to win exactly 95 cents, so I would rule that out.

If we looked at many of these tickets, the average return would be about 95 cents per ticket. That one feels pretty interesting because we're looking at many of these tickets, and so across many of them, you would expect to, on average, get the expected return as your return. And so this is what we are saying here; the average return would be about that. That would be approximately that, so I like that choice. That is a good interpretation of expected value.

And then choice D: if 1,000 people each bought one of these tickets, they'd expect a net gain of about 950 dollars in total. This one is tempting. Instead of net gain, if it just said return, this would make a lot of sense. In fact, it would be completely consistent with choice C. If you have a thousand people, that would be many tickets. And if on average, if their average return is about 95 cents per ticket, then their total return would be about 950.

But they didn't write return here; they wrote net gain. Net gain would be how much you get minus how much you paid. And a thousand people would have to pay. If they each got a ticket, they would pay 2,000. So they would pay 2,000; they would expect a return of 950. Their net gain would actually be negative 1,050 dollars, so we would rule that one out as well.