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"The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes." -- https://en.wikipedia.org/wiki/Expe

"The expected value of a random variable with a finite number of outcomes is a weighted average of all possible outcomes." -- https://en.wikipedia.org/wiki/Expected_value https://en.wikipedia.org/wiki/Expected_value

alexb_ 3y ago
That makes sense, I was always thinking of it as "Given an infinite number of trials..."
- ineptech 3y ago
  Whether/when its better to think in terms of "X has a 37% chance of happening in a single trial" vs "If you ran a lot of trials, X would happen in 37% of them" is kind of a fraught topic that I can't say much about, but you might find this interesting: https://en.wikipedia.org/wiki/Probability_interpretations https://en.wikipedia.org/wiki/Probability_interpretations
- tnecniv 3y ago
  That would be the frequentist interpretation. A Bayesian would say that probability is to be interpreted in belief that an outcome will occur. Neither is really right or wrong, it depends on what you're modeling. If we’re analyzing some kind of heavily repeated task (e.g., a sordid night of us glued to the blackjack table where we play a lot of hands or data transmission over a noisy cable), a frequentist interpretation might feel more sense. However if you’re talking about the probability of a candidate winning an election, you could take a Bayesian view where the probability asserts a confidence in an outcome. A radical frequentist would take umbrage with an event that only happens once. However, I suppose, depending on your election rules and model (e.g., a direct democracy), you could interpret the election winner in a frequentist manner: the probability of winning is the rate at which people vote for the candidate. For a more complicated system I’m not sure the frequentist view is as easily justified. However to answer your question more directly, the expected value is just another name for the average or mean of a random variable. In this case, the variable is your profit. Assume we’re betting a dollar per toss on coin flips and I win if it’s heads (everyone knows heads always wins, right?). The expected value is probability of heads * 1 - probability of tails * 1. If the coin is fair, the probabilities are the same so the expected value is zero. Aside: sequences of random variables that are “fair bets” are called martingales and are incredibly useful. It’s a fair bet because, given all prior knowledge of the value of the variable thus far, the expected value of the next value you witness is the current value of the variable. You could imagine looking at a history of stock values. Given all that information, it’s a martingale (and thus a fair bet) if given that information your expected profit from investing is 0.

alexb_ 3y ago

That makes sense, I was always thinking of it as "Given an infinite number of trials..."

ineptech 3y ago
Whether/when its better to think in terms of "X has a 37% chance of happening in a single trial" vs "If you ran a lot of trials, X would happen in 37% of them" is kind of a fraught topic that I can't say much about, but you might find this interesting: https://en.wikipedia.org/wiki/Probability_interpretations https://en.wikipedia.org/wiki/Probability_interpretations
tnecniv 3y ago
That would be the frequentist interpretation. A Bayesian would say that probability is to be interpreted in belief that an outcome will occur. Neither is really right or wrong, it depends on what you're modeling. If we’re analyzing some kind of heavily repeated task (e.g., a sordid night of us glued to the blackjack table where we play a lot of hands or data transmission over a noisy cable), a frequentist interpretation might feel more sense. However if you’re talking about the probability of a candidate winning an election, you could take a Bayesian view where the probability asserts a confidence in an outcome. A radical frequentist would take umbrage with an event that only happens once. However, I suppose, depending on your election rules and model (e.g., a direct democracy), you could interpret the election winner in a frequentist manner: the probability of winning is the rate at which people vote for the candidate. For a more complicated system I’m not sure the frequentist view is as easily justified. However to answer your question more directly, the expected value is just another name for the average or mean of a random variable. In this case, the variable is your profit. Assume we’re betting a dollar per toss on coin flips and I win if it’s heads (everyone knows heads always wins, right?). The expected value is probability of heads * 1 - probability of tails * 1. If the coin is fair, the probabilities are the same so the expected value is zero. Aside: sequences of random variables that are “fair bets” are called martingales and are incredibly useful. It’s a fair bet because, given all prior knowledge of the value of the variable thus far, the expected value of the next value you witness is the current value of the variable. You could imagine looking at a history of stock values. Given all that information, it’s a martingale (and thus a fair bet) if given that information your expected profit from investing is 0.