EPTCoins tosses - Page 7
| Blogs > Sea_Food |
|
hypercube
Hungary2735 Posts
| ||
|
Cassel_Castle
United States820 Posts
B = HT C = TH D = At least one coin is heads P(A|D) = P(A&D)/P(D) (bayes) =0.25/0.75 =1/3 P(B|D) = P(B&D)/P(D) =0.25/0.75 =1/3 and same for C. P(A&D)=P(A), P(B&D)=P(B), etc. because A, B, and C all imply D. | ||
|
Sea_Food
Finland1612 Posts
On July 19 2011 03:30 Sea_Food wrote: I am going to type claims, and you will just have to type which are the claims that are incorrect. You can explain why if you want. 1. Whenever two coins are tossed there is a 1/2 chance they both land different side up. 2. Whenever two coins are tossed, and you are told one of them landed on heads, you know there is a 2/3 chance the two coins landed different side up. 3. Whenever two coins are tossed, and you are told one of them landed on tails, you know there is a 2/3 chance the two coins landed different side up. 4. Whenever two coins are tossed, and you are told one of them landed on X, you know there is a 2/3 chance the two coins landed different side up. 5. When someone says he is going to toss two coins, but he promises he will tell you on what side one of them landed, you can before the coins are tossed already tell that there is 2/3 chance they will land different side up. 6. Two coins are tossed 1 000 000 times. Every time two tails land you are told one landed tails. Every time two heads land you are told one landed heads. When its 1 tails, one heads, you are told 50% times one landed heads, other 50% times one landed tails. No matter what, every time you are told which side up one of the coins landed, you will say there is 2/3 chance the two coins landed different side up. The rules on what base are you told what coin landed which side are not told to you. 7. Same as 6, but the rules are told to you. Predictions: + Show Spoiler + Unsure about 4, but 5, 6 and 7. But then again im the guy who dosnt even understand 2 or 3. Anyone? Just tell me the incorrect statements please. | ||
Chill
Calgary25977 Posts
| ||
|
Riku
United States1064 Posts
Definitely very interesting! | ||
|
aNDRoM
United States637 Posts
On July 16 2011 02:35 Sea_Food wrote: Your friend tosses two coins, then asks you to guess how the coins landed. You reply that you cannot know. Then your friend reveals that one of the coins he threw landed heads. Now how did the other coin land? Your friend tosses two coins and lay them out on the table, covered (by ESPORTS!): ![[image loading]](http://i.imgur.com/xb1T0.png) He asks you to guess how the coins landed. You reply that you cannot know. Then your friend reveals that one of the coins he threw landed heads by uncovering it: ![[image loading]](http://i.imgur.com/fvpdK.png) or He points to the covered coin and asks how the other coin landed? If your friend actually "reveals" which coin is heads, and asks only what the "other" coin is, then the answer is that the other coin is equally likely to be heads or tails, since any coin, considered individually, has those odds (assuming perfect balance and not landing on its edge, etc). (This is one interpretation, and based on the original phrasing, one could argue that it's the most logical interpretation. I'm working on the other interpretation, the one that involves the "unintuitive" 1/3 heads, 2/3 tails, odds...) | ||
|
Hesmyrr
Canada5776 Posts
On to the topic, yes all seven sentences are valid. Someone point it out if I am wrong though. | ||
|
Chill
Calgary25977 Posts
1. Whenever two coins are tossed there is a 1/2 chance they both land different side up. True.TT TH HT HH 2/4 2. Whenever two coins are tossed, and you are told one of them landed on heads, you know there is a 2/3 chance the two coins landed different side up. True.TH HT HH 2/3 3. Whenever two coins are tossed, and you are told one of them landed on tails, you know there is a 2/3 chance the two coins landed different side up. True.TT TH HT 2/3 4. Whenever two coins are tossed, and you are told one of them landed on X, you know there is a 2/3 chance the two coins landed different side up. True.XX XY YX 2/3 5. When someone says he is going to toss two coins, but he promises he will tell you on what side one of them landed, you can before the coins are tossed already tell that there is 2/3 chance they will land different side up. False.All options are still possible. If it lands TT, he will say "One of them is tails." If it lands HH, he will say "One of them is heads." Because he gets to choose, we can't eliminate any options. TT TH HT HH 2/4 6. Two coins are tossed 1 000 000 times. Every time two tails land you are told one landed tails. Every time two heads land you are told one landed heads. When its 1 tails, one heads, you are told 50% times one landed heads, other 50% times one landed tails. No matter what, every time you are told which side up one of the coins landed, you will say there is 2/3 chance the two coins landed different side up. The rules on what base are you told what coin landed which side are not told to you. True. This is actually the same as question 3 and 4 but you've worded it strangely.7. Same as 6, but the rules are told to you. True. Because the rules are the same as question 3 and 4 (fair and random), knowing the rules doesn't change anything. | ||
|
Chill
Calgary25977 Posts
On August 11 2011 05:56 Hesmyrr wrote: Wait, the OP actually managed to rhyme "coin" and "toss" together? O.O On to the topic, yes all seven sentences are valid. Someone point it out if I am wrong though. No, it was terrible but I opened it because he put in the effort. | ||
|
Chill
Calgary25977 Posts
| ||
|
aNDRoM
United States637 Posts
However, the other interpretation of the question is that he's asking what the chances are that the other coin landed heads (therefore, since he already said he has one heads, he would end up with two heads), and what are the chances that the other coin landed tails (therefore he has a heads and a tails). --- Let's play three card monte, but instead of a red queen, we'll use a heads up coin, and we'll use more ESPORTS to cover the coins. I put three coins on a table, one heads, two tails: + Show Spoiler + ![[image loading]](http://i.imgur.com/jgPw3.png) I cover them up and rearrange them. + Show Spoiler + Can we all agree that if I point at a covered coin, that it no longer has an equal chance of being heads or tails? You saw one heads and two tails and all I did was cover them up, so even though they're coins, they're not being reflipped, so if you pick one of the three at random, you should have a 1/3 chance of picking the heads and a 2/3 chance of picking the tails. This is the key to the whole thing. In the above situation, you're no longer calculating whether a coin has landed heads or tails. You're picking from a fixed quantity of heads and tails. This may seem contrived, but let's keep going. On another table, I have three coins, all heads: + Show Spoiler + I then put one of my three covered coins next to each of the heads, forming three rows of two coins each: + Show Spoiler + Even if I don't know what each covered coin is, I know one row has two heads, and the other two rows have a heads and a tails each. I then add a row of two coins, both tails: + Show Spoiler + The four rows represent all the possible ways two coins flipped together could end up: HH, HT, TH, TT. Yes, since I had all heads on the left to start, if I flipped over all the coins right now, there would be a "repeat" row, like this: + Show Spoiler + But that doesn't mean you can discount that there's still four separate possibilities for how two coins could land, even with the "repeat" row. The above is a reverse of how to calculate the 1/3 heads, 2/3 tails odds. You start with the four possible ways two coins could land together, eliminate the TT (no pun intended) since you know at least one of the coins is heads, and see what the odds are that both coins are heads or that the other coin is tails. | ||
|
aNDRoM
United States637 Posts
On August 11 2011 06:39 Chill wrote: To summarize: You are still trying to prove yourself right by coming up with weird scenarios (#5, #6, #7). You can't use probability to predict the future of a truly random event (#5). Despite what your friend says he is going to do, a coin flip is still 50%. Isn't the OP on the "50%" camp and not on the 1/3, 2/3 side? | ||
|
Chill
Calgary25977 Posts
On August 11 2011 06:47 aNDRoM wrote: Isn't the OP on the "50%" camp and not on the 1/3, 2/3 side? Yes. And he's invented scenario #5 to show how you can affect the future with your words. That's my point. Scenario #5, where your friend says "Okay in the future I'll do this" is different than the rest because he hasn't affected anything by saying that. So that case is 1/2. Everything else is 2/3. | ||
|
aNDRoM
United States637 Posts
On August 11 2011 06:51 Chill wrote: Yes. And he's invented scenario #5 to show how you can affect the future with your words. That's my point. Scenario #5, where your friend says "Okay in the future I'll do this" is different than the rest because he hasn't affected anything by saying that. So that case is 1/2. Everything else is 2/3. Sorry, got it. (I think the question is vague enough for interpreting it the way he wants to understand it, which is an issue with a lot of math probability questions, as shown in the boy/girl wikipedia entry.) | ||
|
Slithe
United States985 Posts
You friend flips two coins, and asks, "What is the probability of the each of the possible results?" Answer: HH = 1/4 HT = 1/4 TH = 1/4 TT = 1/4 Now the guy says "I have checked the coins, and I can tell you that the result is not TT. Now, what is the probability of the coin results?" Answer: HH = 1/3 HT = 1/3 TH = 1/3 TT = 0 Now the guy points to the first coin and asks, "What is the probability that this coin is heads?" Answer: HH + HT = 2/3 This is just rewording the same thing again, but maybe this progression will make more sense to you. | ||
|
numLoCK
Canada1416 Posts
Lets say I flip two coins. I check the first coin, and see it is heads. I now know that the first coin landed heads. That means that there is a 50% chance that the second coin landed heads, and a 50% chance that it landed tails. So if I guess, I have a 50/50 chance of getting it right. If I then tell my friend that at least one of the two coins landed heads (a true statement), according to the results of this thread, he guesses tails at 66% accuracy. However, there is no way that his incomplete information gives him better odds than me. In fact, I am certain I must be going about this the wrong way, but just can't see it. So please tell me what I'm doing wrong. edit: nvm, I get it. lol I was dumb for a moment there  | ||
|
Chill
Calgary25977 Posts
| ||
|
numLoCK
Canada1416 Posts
So if the situation is this: I flip 2 coins 100 times. In each case, I look at the first one and guess that the other coin will be whichever the first one wasn't. Because I know the first coin is X, then not X will have 50% likely and I will be correct approx. 50 times. Also in each case, I tell my friend that at least one coin is X, so he guesses that the other coin will be not X. By the logic of the thread, he should be correct at a 66% rate, but he will in fact be correct approx. 50 times (same as me, we have the same guess). Explain to me why this is, and why it does not show that the percentage should in fact be 50, not 66? Actually, I guess it is just that the way you guys are doing it, out of the 100 cases we would only play those that landed HT, TH, HH, and we wouldn't play TT at all. By playing all four options and switching between H or T for X we ruin it. | ||
|
Chill
Calgary25977 Posts
| ||
|
aNDRoM
United States637 Posts
For the first scenario, you're not excluding any of the 100 times the coins are flipped. Once again, assuming perfect coins: Coin One is X on 50 flips. You guess "not X" on those flips. Coin 2 will be X on 25 of those 50 flips and "not X" on the other 25 flips. You'll be correct 50% of those 50 flips. Coin One is "not X" on the other 50 flips. You guess X on those flips. Coin 2 will be "not X" on 25 of those 50 flips and X on the other 25 flips. You'll be correct 50% of those 50 flips. So you'll be 50% correct on all 100 flips. Using the result of the first coin to guess the opposite result for the second coin is as arbitrary a way of choosing heads or tails as flipping a coin... Wait. Edit: So Chill, do you know if the OP has come back and been convinced by any of the new explanations? | ||
| ||
