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I really should be able to do this, but my GCSE maths is failing me.
My boy wants to run a stall at the school fete where people try and score football penalties. We had a practice run and i was scoring 25% of the time (it's a small goal!). I'm now trying to work out, based on that success rate, the proportions of people that would score, 0 out of 3, 1 out of 3, 2 out of 3 and 3 out of 3, so we can judge appropriate prizes for each.
Help!
Simplistically it's 0.25^3 (1.56%) for all 3.
1 in 3 is 0.25*0.75*0.75 *3 is 42%
2 in 3 is 0.25*0.25*0.75 *3 is 14%
But that assumes they're independent outcomes and no skill involved.
If the odds of a random person scoring the first shot are 25%, then the odds of that same person scoring a second are a lot higher. E.g. 4 people turn up, 3 of them are your average middle-aged dads, the 4th is Beckham. You're still correct that 25% of the first shots go in. And probably the 2nd and 3rd's. But they're not evenly distributed.
I think, but would appreciate confirmation:
P(0 goals) = 27/64
P(1 goal) = 27/64
P(2 goals) = 9/64
P(3 goals) = 1/64
Hope I’ve not messed up there. The main presumption being everybody kicks like you.
Damn, sorry just what thisisnotaspoon said 😀
i was scoring 25% of the time
Yes, but what will the average customer score? Are you average, better than average, worse than average? If you're just average, how much better is a good goal kicker? If a really good goalkicker just happens to turn up and can kick 90%, then they will score 3 out of 3 more than 70% of the time. That would bankrupt you unless you cut off anyone who wins the big prize.
Yes, but what will the average customer score? Are you average, better than average, worse than average? If you’re just average, how much better is a good goal kicker? If a really good goalkicker just happens to turn up and can kick 90%, then they will score 3 out of 3 more than 70% of the time. That would bankrupt you unless you cut off anyone who wins the big prize.
Use three penalty spots progressively further back. Every goal you score all afternoon puts you one spot back. First spot is quite close so everyone has a chance of a very low value prize at least once. 2nd spot is at a normal distance and the third far one makes it quite tough. That make the jackpot for the first time players (one from each spot) hard to achieve but if Beckham rocks up on his 2nd go he'd need to score all three from the far spot significantly reducing his chances.
Reminds me of the story my Dad told of when they first went to France. There was an amusement arcade at the beach featuring a stall where you had to throw darts at balloons to win a bottle of wine. They don't play darts in France, so noone was any good and the stall owner was able to make a living. My Dad on the other hand was a handy pub darts player and popped every balloon he aimed at. They used to go there to get their evening bottle of wine every night until he got banned.
A nice problem is the probability of scoring at least one goal. This is the coupon collectors problem.
probability of scoring = 1/4
probability of NOT scoring = 3/4
probability of NEVER scoring (3/4)^3
probability of not never scoring (I.e., scoring at least once) is 1 - (3/4)^3 or 58%.
Those are pretty good odds. Now turn that into a hole in one in a major golf championship where p is small (say 1/10000 instead of 1/4) but N is huge, 156 players, or about 16848 holes played if half play four rounds and half play two. The probability of at least one hole in one is about 81%%. At 1/1000 it’s a near certainty. Multiplicity is a marvellous thing and you won’t get decent odds on a hole in one any more.
https://www.gambling.com/uk/news/the-hole-in-one-gang-how-golf-bettors-beat-the-bookies-2232000
TBH I thought the probability of me reading an interesting story about golf was a million to one but that is an interesting story.
Thanks for the help folks.
I've plugged all the numbers into a spreadsheet so he can model the financial outcome if you change the various parameters. Obviously the 25% success rate on each kick is the unpredictable variable, but we can at least play with the figures and see what safety net we have if that's an under estimate.
<p>A bit OTT for your 3-shot penalty simulation work, but this piece on Markov models in football might be a fun follow up. https://statsbomb.com/articles/soccer/attacking-contributions-markov-models-for-football/ </p>