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Interpreting expected value

We can interpret expected value as a long term average outcome. This example looks at expected value in the context of a lottery ticket. Created by Sal Khan.

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  • leaf yellow style avatar for user John
    At , how is he getting the probability the probability of wining to be instead of ?
    (4 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      The question states that the odds of winning are 1∶50,
      which means that if 𝑝 is the probability of winning, then the probability of losing is 50𝑝.

      Since winning or losing are the only possible outcomes when playing the game,
      we know that 𝑝 + 50𝑝 = 1
      ⇒ 51𝑝 =1
      ⇒ 𝑝 = 1∕51
      (12 votes)
  • blobby green style avatar for user 春霞 尤
    I think the correct conversion of to fraction is 1/50.
    (1 vote)
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  • leaf green style avatar for user Erik Suárez Cosío
    My understanding of expected value from previous videos, is that the cost was included when calculating the weighted sum. So if the expected return/value is 0.95 dollars per ticket, according to my understanding and chatgpt, this means including the 2 dollar cost.

    In other Khan Academy videos, Sal would include the cost in the operation for the expected value. E.g. p_1 (prize money from outcome 1 - cost for ticket) + p_2 (prize money from outcome 2 - cost for ticket)...

    It seems to me that there was an inconsistency in this video. Also, bing ai says that 1 : 50 means a probability of 1/50 and not 1/51.
    (0 votes)
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    • leaf green style avatar for user kubleeka
      "Expected return" and "expected value" are not the same thing. "Return" here means the simple amount of money that you (expect to) win, regardless of what you paid. The expected return will always be positive for a typical lottery, because you will either win something, or win nothing, but you are never penalized further for losing.

      The value takes into account the cost of the ticket, which drives it negative. The expected value is the expected return minus the expected cost.

      Also, means that for every win, there are 50 losses. That is, there is one win out of a total of
      1 win + 50 losses = 51 trials
      or 1/51. Bing AI is wrong.
      (4 votes)
  • piceratops seed style avatar for user pkannan.wiz
    If anyone is curious on how much you would win.
    Suppose you win $x. Then it follows that 1/50 * x - 2 = 0.95. Solve to get x = 147.50.
    (0 votes)
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  • blobby green style avatar for user KevinS
    yo chat is this real?
    (0 votes)
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Video transcript

- [Instructor] We're told a certain lottery ticket costs $2 and the back of the ticket says, "The overall odds of winning a prize with this ticket are one to 50, and the expected return for this ticket is $0.95." Which interpretations of the expected value are correct? Choose all answers that apply. Pause this video, have a go at that. 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 one to 50, that means that the probability of winning is one 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 $0.95. 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 $0.95. 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 $0.95. So it's actually maybe even impossible to win exactly $0.95. So I would rule that out. If we looked at many of these tickets, the average return would be about $0.95 per ticket. That one feels pretty interesting, 'cause 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 seeing here. The average return would be about that. It 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 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 1,000 people, that would be many tickets, and if on average, if their average return is about $0.95 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 1,000 people would have to pay, if they each got a ticket, 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. So we would rule that one out as well.