What are the odds of not being selected 2 years in a row when you have a 71% chance of winning each year. We have an annual lottery for a golf tournament. 130 registrants, 92 are picked each year.
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1$\begingroup$ The probability would be $\left( \frac{38}{130} \right)^2$. $\endgroup$– AmateurCommented Apr 6, 2014 at 1:28
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$\begingroup$ Thanks! Does that equate to about an 8.5% chance of that happening? Would I cube it if I was looking for the chance of not being selected 3 years in row, as in about 2.5%? Much appreciated. $\endgroup$– user140742Commented Apr 6, 2014 at 1:35
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$\begingroup$ Yes, it equates to about an $8.5$% chance of that happening. You would indeed cube it if you were looking for the chance of not being selected $3$ years in a row and that would give about a $2.5$% chance. $\endgroup$– AmateurCommented Apr 6, 2014 at 1:37
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Probability of two independent events is the product P(A and B) = P(A) x P(B), assuming that A and B are independent. In this case the odds of not being selected on any two years (in a row or not) is .29 x .29 = .0841, which is about 8.4%. I don't think you should suspect a conspiracy. Consider that 38 are not picked on the first year, and of those 38, 3 will not be picked on the second year either. Of those 3 it is likely that one of them will post about it on stack exchange ;-)
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$\begingroup$ Great answer and perfect. No conspiracy! :) I am actually the one that has to explain to those that are not selected that the math is ultimately in their favor. Thank you very much. $\endgroup$ Commented Apr 6, 2014 at 1:39
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$\begingroup$ It's still pretty bad luck, you would expect (on average) only about $11$ people to have such bad luck. $\endgroup$– JaredCommented Apr 6, 2014 at 1:50
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$\begingroup$ Wouldn't the "odds" be $\frac{.29^2}{1 - .29^2}$ and not $.29^2$? $\endgroup$– GridCommented Apr 6, 2014 at 3:54
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$\begingroup$ @user140742 ~~ I disagree that it is "ultimately in their favor". In your current system, there is a 40% chance that someone will lose four years in a row! That's harsh. If I were you, I'd change the lottery so that if a person loses one year, then he or she is guaranteed to win on the next year. That's sensible, fair, and much more polite. $\endgroup$ Commented Apr 10, 2014 at 21:47