If I win the bet I get 200 dollars and my winning probability is 0.1 and if I lose I give 20 dollars. Should I take the bet?
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3$\begingroup$ It depends entirely on how you value the outcomes, which relates to how you value money and risk. $\endgroup$– Glen_bCommented Oct 6, 2013 at 7:28
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1$\begingroup$ I wouldn't. Winning \$200 won't change my life, nor will losing \$20. So why waste my time on this? $\endgroup$– Harvey MotulskyCommented Oct 6, 2013 at 15:27
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1 Answer
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It depends on many factors. First - are you taking the bet once or as long as you want?
- If you take it just once then the answer is no, as there is much higher probability of losing money then winning(even though the expected value is positive)
- If you can take this bet for as long as you want, it is a good bet, as with expected value of income equal to 2, we could expect that if we play long enough - we will gain more and more money
In general the problem here is weighting the expected outcome (expected value) and the risk (variance) of the process. In "real life", where all resources are limited, the risk factor may be much bigger then the one coming from expected outcome (which is just a theoretical entity). As such there is no good answer for "real world" bet, only for theoretical betting games, where there exists infinity
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$\begingroup$ Thanks, and if suppose I had to give 10 dollars, then I should take it right? This means that if probability of me winning some lottery is 0.00001 but I get 1 million for it, and the lottery ticket is 5 dolars, then I should buy the lottery? $\endgroup$– PrashantCommented Oct 6, 2013 at 7:36
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1$\begingroup$ If you plan to do this only once then it is not so easy. It depends on what is your "aim", as there is still bigger chance of losing some money, no matter how big is a possible prize. Consider something like this: you have 0.00000000000000000001 chance of winning power of becoming a god, and 0.99999999999999999999 that you will lose your arm. Even though becoming a god should have nearly infinite "reward", and so expected value of this bet is positive it does not mean that a rational person would go for it, as with enormous chance you will lose an arm $\endgroup$– lejlotCommented Oct 6, 2013 at 7:49
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$\begingroup$ In case of a lottery: if you can but as much tickets as you want (you have enough money), the lottery is fair (random) and only you can win (there is no top-winners splitting policy) then once the expected value is bigger then zero, you should buy ALL possible tickets (in case of some number guessing lottery) and you can rest assured that you will win. In more randomized lotteries it is still statistically better to buy as much as you only can (if above conditions hold and that there is true randomization in bought tickets) but with just a probabilistic "guarantees". $\endgroup$– lejlotCommented Oct 6, 2013 at 7:54
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$\begingroup$ Am I completely off or isn't the expected income 2? $0.1 \cdot 200 + 0.9 \cdot -20 = 2$. Then I'll think it is a quite good bet, though, depending on the factors Glen_b mentions. $\endgroup$ Commented Oct 6, 2013 at 21:06
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2$\begingroup$ This answer is incorrect for two reasons. First, the expected income is $200*0.1-20*0.9=2$, so this is a positive expectation bet. Second, if it were not a positive expectation bet, then it would not be a good idea to take it even if you could do it as many times as you want. Yes, you would eventually end up with a positive amount of money, but the expected value of the amount of time until this happens is infinite. Moreover, if you have a finite bankroll, and you play long enough, then with probability $1$ you will eventually go broke. $\endgroup$– mprCommented Oct 6, 2013 at 22:15