Coin Flip Odds

BillPorter said:
Patrick, Wikipedia offers a nice analysis of this problem: http://en.wikipedia.org/wiki/Wine/Water_mixing_problem

As it turns out, it is irrelevant whether you mix well or poorly when you add one liquid to the other.
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Yes, because to make room in the spoon for whatever amount of Kahlua you return to the glass of milk, you must be leaving that same amount of milk in the Kahlua. If I had realized this in the first place I wouldn't have needed a graph!

pj
chgo
 
StevenPWaldon said:
Has anyone out there lost more than 7 consecutive coin flips?
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Here is one for you....

A few week ago 2 friends and myself were going to play 3 way snooker. We decided to flip to determine the order. Odd person out would go first.. then a simple flip to determine second... well...

7 times in a row we either all flipped heads or all flipped tails.

The odds of doing that are 0.25^7 = 0.000061

or 0.0061%

or 6.1 times in 100,000 tries
 
eastcoast_chris said:
Here is one for you....

A few week ago 2 friends and myself were going to play 3 way snooker. We decided to flip to determine the order. Odd person out would go first.. then a simple flip to determine second... well...

7 times in a row we either all flipped heads or all flipped tails.

The odds of doing that are 0.25^7 = 0.000061

or 0.0061%

or 6.1 times in 100,000 tries
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Since it wouldn't have mattered which of your three coins fell first, second, or third, you would be dealing with combinations, not permutations. As I figure it, there are only 4 combinations of three coins as follows: 1) 3 heads, 2) 2 heads and a tail, 3) 1 head and two tails, and 4) three tails. Notice that two out of four of these combinations involve either all heads or all tails. So you would expect to get all heads OR all tails half the time. Getting three heads or three tails 7 times in a row boils down to getting a head (or tail) 7 times in a row. So the odds would be 1 out of 128 or about .78%. So if you did this every day for a year, I'd expect that it would happen 2 or 3 times during the year.
 
BillPorter said:
Since it wouldn't have mattered which of your three coins fell first, second, or third, you would be dealing with combinations, not permutations. ... (50-50 chance)
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Perhaps we can arrange a field trial. I'll take the 3:1 position.;)
 
BillPorter said:
Since it wouldn't have mattered which of your three coins fell first, second, or third, you would be dealing with combinations, not permutations. As I figure it, there are only 4 combinations of three coins as follows: 1) 3 heads, 2) 2 heads and a tail, 3) 1 head and two tails, and 4) three tails. Notice that two out of four of these combinations involve either all heads or all tails. So you would expect to get all heads OR all tails half the time. Getting three heads or three tails 7 times in a row boils down to getting a head (or tail) 7 times in a row. So the odds would be 1 out of 128 or about .78%. So if you did this every day for a year, I'd expect that it would happen 2 or 3 times during the year.
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I think you're wrong about combinations vs. permutations when it comes to the probability of 3 coins coming up all heads or all tails. Disregarding combination vs. permutations, we can cut the confusion this way: the probability of 3 heads = .5^3 = .125. The probability of 3 tails is the same (unless I'm thinking according to the mistaken paradigm that the universe exists outside my own navel (far out, man)). Since 3 heads and 3 tails are mutually exclusive outcomes, the probability of 3 heads OR 3 tails = .125 + .125 = .25, which is what the other poster said.

-Andrew
 
BillPorter said:
Since it wouldn't have mattered which of your three coins fell first, second, or third, you would be dealing with combinations, not permutations. As I figure it, there are only 4 combinations of three coins as follows: 1) 3 heads, 2) 2 heads and a tail, 3) 1 head and two tails, and 4) three tails. Notice that two out of four of these combinations involve either all heads or all tails. So you would expect to get all heads OR all tails half the time. Getting three heads or three tails 7 times in a row boils down to getting a head (or tail) 7 times in a row. So the odds would be 1 out of 128 or about .78%. So if you did this every day for a year, I'd expect that it would happen 2 or 3 times during the year.
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Not quite: Yes, there are only 4 possible combinations, but the permutations do matter ... there are 8 ways to get them... of which only 2 are all heads or all tails.... producing a 25% probability of doing this 7 times in a row.... think of it simply by asking.... what are the odds of flipping 2 coins and they be the same? 50% of course... so after 2 coins, 50% of the time you've already failed. and of the 50% when you haven't failed, there is a 50% chance you will fail on the next (3rd) flip.... giving you a 25% chance of success.

I wrote a quick Java program to prove this to myself and in 100,000 samples 24,986 we all heads or all tails. =~ 25%

0.25^7 is the probability of doing this 7 times in a row... which = 1 in 16384
 
BillPorter said:
Since it wouldn't have mattered which of your three coins fell first, second, or third, you would be dealing with combinations, not permutations. As I figure it, there are only 4 combinations of three coins as follows: 1) 3 heads, 2) 2 heads and a tail, 3) 1 head and two tails, and 4) three tails. Notice that two out of four of these combinations involve either all heads or all tails. So you would expect to get all heads OR all tails half the time. Getting three heads or three tails 7 times in a row boils down to getting a head (or tail) 7 times in a row. So the odds would be 1 out of 128 or about .78%. So if you did this every day for a year, I'd expect that it would happen 2 or 3 times during the year.
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Nope Bill.......faulty logic.......eastcoast_chris is absolutely correct.

You are correct that there are only 4 different 'results' when you summarise them like that but in fact there are 8 ways in which the three coins can land in order to achieve those 4 different results (av8forfun, are you watching as this loosely relates to our earlier discussion re probabilities?:) )

It is the existence of 8 different ways in which the three coins can land to result in those 4 different combinations (as you put it) which matter as far as probability goes.The 8 combinations in which the three coins can land are as follows......two of the combinations involve all being heads and all being tails, three combinations involve two tails and a head and the remaining three combinations involve two heads and a tail.

The chances of all three coins being the same (ie all heads or all tails) in any one toss of all three can therefore be expressed as odds of 6/2 (six in which all three coins aren't the same and two in which they are all the same), which we would more normally express as 3/1 against. For the record the odds against all three coins coming up as heads are 7/1 and as all tails are 7/1.

The odds against all three coins coming up either all heads or all tails 7 times in a row are exactly 16,383/1, the same as backing 7 winning horses in an accumulator at odds of 3/1 each which is what eastcoast_chris expressed in a more scolarly manner than me:)

Hope this helps clear it up. If you are still having difficulty seeing it try writing down three columns and put all the possible ways the coins can land under Coin1, Coin 2 and Coin 3 columns and then add them up.
 
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coin flip odds

RE the coin flip discussion: EVERY time you flip a coin, the odds are 50:50. (I have never seen one land on it's side.) The only way you can obtain an edge, and it is a small one, is to have your opponent call the flip. The reasoning behind this is that people call "heads" about 70% of the time. You don't have to be an Einstein to see how this works. However, if anyone wants to bet that they can call the flip correctly eight times in a row with a standard US 25cent piece, I will gladly give them three to one. The main point, which has already been made many times, is that the coin has no memory, the odds on the first flip are the same as the odds on the thousandth.
 
The only way you can obtain an edge, and it is a small one, is to have your opponent call the flip. The reasoning behind this is that people call "heads" about 70% of the time. You don't have to be an Einstein to see how this works.
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As somebody pointed out earlier, if you call heads 70 percent of the time and tails 30 percent of the time, you'll still win about half of each. If you call heads 100 percent of the time, you'll win about half. No matter how you call them, you'll win about half in the long run.

pj
chgo
 
But what if someone actually practices filipping a coin and gets good at flipping it so it makes an odd or even number of revolutions in the air, thus increasing the odds of getting a favorable flip?
 
memikey said:
Nope Bill.......faulty logic.......eastcoast_chris is absolutely correct.

You are correct that there are only 4 different 'results' when you summarise them like that but in fact there are 8 ways in which the three coins can land in order to achieve those 4 different results (av8forfun, are you watching as this loosely relates to our earlier discussion re probabilities?:) )

It is the existence of 8 different ways in which the three coins can land to result in those 4 different combinations (as you put it) which matter as far as probability goes.The 8 combinations in which the three coins can land are as follows......two of the combinations involve all being heads and all being tails, three combinations involve two tails and a head and the remaining three combinations involve two heads and a tail.

The chances of all three coins being the same (ie all heads or all tails) in any one toss of all three can therefore be expressed as odds of 6/2 (six in which all three coins aren't the same and two in which they are all the same), which we would more normally express as 3/1 against. For the record the odds against all three coins coming up as heads are 7/1 and as all tails are 7/1.

The odds against all three coins coming up either all heads or all tails 7 times in a row are exactly 16,383/1, the same as backing 7 winning horses in an accumulator at odds of 3/1 each which is what eastcoast_chris expressed in a more scolarly manner than me:)

Hope this helps clear it up. If you are still having difficulty seeing it try writing down three columns and put all the possible ways the coins can land under Coin1, Coin 2 and Coin 3 columns and the add them up.
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You are absolutely correct. Guess I slipped a cognitive gear!:D Maybe I just tried to come up with something that looked plausible if you didn't look too closely.
 
eastcoast_chris said:
Not quite: Yes, there are only 4 possible combinations, but the permutations do matter ... there are 8 ways to get them... of which only 2 are all heads or all tails.... producing a 25% probability of doing this 7 times in a row.... think of it simply by asking.... what are the odds of flipping 2 coins and they be the same? 50% of course... so after 2 coins, 50% of the time you've already failed. and of the 50% when you haven't failed, there is a 50% chance you will fail on the next (3rd) flip.... giving you a 25% chance of success.

I wrote a quick Java program to prove this to myself and in 100,000 samples 24,986 we all heads or all tails. =~ 25%

0.25^7 is the probability of doing this 7 times in a row... which = 1 in 16384
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I completely agree! Thanks for the correction.:)
 
MBTaylor said:
But what if someone actually practices flipping a coin and gets good at flipping it so it makes an odd or even number of revolutions in the air, thus increasing the odds of getting a favorable flip?
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I know someone who claims to have made a lot of money flipping coins. There are lots of ways to improve your odds. Here's one: nick the edge of the coin on the head side. Flip it and have your opponent call it in the air and catch it between your fingers and against the back of your hand, noting which finger has the nick against it. Shift your right hand to let the correct side of the coin fall against the back of your left hand. I suppose this doesn't really require a nick if you have quick eyes. I certainly have never used this as I'm not coordinated or large enough to pull it off, but I heard about this method about 40 years ago.
 
an easy fix

Just do what Spanish Mike Lebron does and call "Hails".........
 
Bob Jewett said:
I know someone who claims to have made a lot of money flipping coins. There are lots of ways to improve your odds. Here's one: nick the edge of the coin on the head side. Flip it and have your opponent call it in the air and catch it between your fingers and against the back of your hand, noting which finger has the nick against it. Shift your right hand to let the correct side of the coin fall against the back of your left hand. I suppose this doesn't really require a nick if you have quick eyes. I certainly have never used this as I'm not coordinated or large enough to pull it off, but I heard about this method about 40 years ago.
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I learned that in a magician's book but it had nothing to do with a nick in this case. I can easily use a borrowed quarter and rub right before putting on my hand after I catch the flip and simply feel for the smoothness of the heads side. Simple sleight of hand.

It would take a real sucker to not insist that it land on the table though.
 
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