Skuxxgolfer
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What are some hard past papers?
Look at BOS Trials for hard papers.What are some hard past papers?
Is the answer 1/2? You can tell by symmetry, but it can be proven using binomial probability and binomial theorem as well.
It shouldn't be exactly 0.5, since they flip a different number of coins.Is the answer 1/2? You can tell by symmetry, but it can be proven using binomial probability and binomial theorem as well.
There are two cases to consider: Ben obtains more tails than Amy, or Ben obtains more heads than Amy. Since it is a fair coin, these two cases have the same probability, by symmetry. That means that the probability that Ben obtains more heads is exactly a half. This can also be proven using binomial theorem, which is attached below (This way is unnecessarily long and complex, I just wanted an extra proof to back my case up. Note I also may have made a mistake in it, I just quickly wrote it up tonight).It shouldn't be exactly 0.5, since they flip a different number of coins.
Hm, but if we take a more extreme example: If Amy flips 1 coin and Ben flips 100 coins, then the probability of Ben flipping more heads than Amy would be almost 100% wouldn't it? Or is this 50% result only true for n, n+1 flips? If so, then fair enough, I will take your word for it (it's too late for binomial theorem, not to mention I don't like to do math in my spare time lol).There are two cases to consider: Ben obtains more tails than Amy, or Ben obtains more heads than Amy. Since it is a fair coin, these two cases have the same probability, by symmetry. That means that the probability that Ben obtains more heads is exactly a half. This can also be proven using binomial theorem, which is attached below (This way is unnecessarily long and complex, I just wanted an extra proof to back my case up. Note I also may have made a mistake in it, I just quickly wrote it up tonight).
Yeah I believe it only works for n, (n+1). Because if you take, for example n, (n+2), then it is possible for Ben to have more tails and more heads that Amy, adding a complete new case. Since there are more cases to consider here than just having more heads or more tails, the probability will not be exactly a half. Same thing applies to n+3, n+4 etc.Hm, but if we take a more extreme example: If Amy flips 1 coin and Ben flips 100 coins, then the probability of Ben flipping more heads than Amy would be almost 100% wouldn't it? Or is this 50% result only true for n, n+1 flips?
Yeh that's what I was originally thinking, I just extrapolated (albeit incorrectly) based on the extreme scenario. Good problem nonetheless, shows that it's counterintuitive (or intuitive?) for n, n+1.Yeah I believe it only works for n, (n+1). Because if you take, for example n, (n+2), then it is possible for Ben to have more tails and more heads that Amy, if n is even, as he could have n/2+1 for both, or otherwise he could have the same amount of tails./heads and more heads/tails. Since there are more cases to consider here than just having more heads or more tails, the probability will not be exactly a half. Same thing applies to n+3, n+4 etc.
Yeah I didn't think it was gonna be 1/2 initially either. I ended up coming up with that evil binomial theorem solution,which resulted in 1/2, and then figured out why it worked from there. Now I have wasted my Tuesday night lol.Yeh that's what I was originally thinking, I just extrapolated (albeit incorrectly) based on the extreme scenario. Good problem nonetheless, shows that it's counterintuitive (or intuitive?) for n, n+1.
Yeh that's what I was originally thinking, I just extrapolated (albeit incorrectly) based on the extreme scenario. Good problem nonetheless, shows that it's counterintuitive (or intuitive?) for n, n+1.
I thought this was a very nice problem outlining the utility of symmetry:Yeah I didn't think it was gonna be 1/2 initially either. I ended up coming up with that evil binomial theorem solution,which resulted in 1/2, and then figured out why it worked from there. Now I have wasted my Tuesday night lol.
Your symmetry argument unfortunately fails, as pointed out by blyatman.There are two cases to consider: Ben obtains more tails than Amy, or Ben obtains more heads than Amy. Since it is a fair coin, these two cases have the same probability, by symmetry. That means that the probability that Ben obtains more heads is exactly a half. This can also be proven using binomial theorem, which is attached below (This way is unnecessarily long and complex, I just wanted an extra proof to back my case up. Note I also may have made a mistake in it, I just quickly wrote it up tonight).
Wait sorry why does this argument fail? There are only two possible cases: Ben flips more heads than Amy or Ben flips more tails than Amy, and the 2nd case obviously includes the circumstances where Ben gets the same amount of heads as Amy, as he would have to have more tails then. Then since heads and tails have the same probability, the two cases would have to be equal probability, making the probability 1/2. True straight up saying it is symmetrical is flawed, but I think this way makes sense, unless I'm missing something crucial lol.Your symmetry argument unfortunately fails, as pointed out by blyatman.
I gave this prob to my year's 4u class and most agreed this answer made sense at an intuitive level, but proving it rigorously was much more of a struggle.
No, you can have the case where they flip the same number of Heads.Wait sorry why does this argument fail? There are only two possible cases: Ben flips more heads than Amy or Ben flips more tails than Amy, and the 2nd case obviously includes the circumstances where Ben gets the same amount of heads as Amy, as he would have to have more tails then. Then since heads and tails have the same probability, the two cases would have to be equal probability, making the probability 1/2. True straight up saying it is symmetrical is flawed, but I think this way makes sense, unless I'm missing something crucial lol.
But if they flip the same number of heads, then Ben will flip more tails, since he flips n+1 coins and Amy only flips n, therefore putting it in the 2nd case of Ben flipping more tails than Amy.No, you can have the case where they flip the same number of Heads.