I mentioned to a colleague that I’d bought two lottery tickets for last weeks Euromillions draw instead of my usual 1 to double my chance at winning. He said “Yeah, that’s not how it works.” I’m sure he is right – but why?
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I’d be willing to bet that your colleague is confusing the probability of betting on one event, with the probability of betting on multiple independent events.
Stealing someone elses example from elsewhere in the comments, but let’s imagine you have a wheel split into 5 segments, and you take bets on which segment a marble will land on.
Assuming that it’s truly random, the probability of any one segment being the winner is 20%, so betting on two segments would give you a 40% chance of winning.
But, if you bet on one segment in two independent rounds, your chances are not 40%. Your chances of not winning are 80% (0.8) so your chances of not winning over two rounds is 0.8*0.8 = 0.64 – so you have a 64% chance of not winning and a 36% chance of winning.
If you played the game 5 times, you’d only have a 67% probability of getting a win (probability of the event not occuring is 0.8, so 0.8*0.8*0.8*0.8*0.8 = 0.32768 – round it up to 0.33for simplicity).
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