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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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• At , how is he getting the probability the probability of wining to be instead of ?
• 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
• 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.
• "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.
• I think the correct conversion of to fraction is 1/50.
(1 vote)
• 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.