Well, 14 of you voted. 11 got it wrong, and only 2 got it right (with 1 person not wanting to guess).
The answer is actually that Tails, Heads, Heads comes up with fewer tosses on average. The reasoning is quite easy, but first you need to understand what the question is asking. The question wasn't "Which result requires fewer tosses if you only have 3 tosses" as some people seem to have thought. In that case, the 11 would be correct. If you have only 3 tosses, then all combinations of the 3 tosses (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT) are equally likely of coming up, so THH and THT both have an equal chance (1/8th) of coming up.
However, the question was asking on average, which pattern took the fewest tosses to appear. That means, that if you toss three coins and get HTH, you haven't got either pattern yet and must continue flipping. The same goes for the other 5 patterns that don't equal either THH or THT.
Since we are moving from left to right, we should be able to deduce that both patterns start with TH. This is the common factor between them, and it turns out to be very important. To answer the question, it is necessary to see what happens when each pattern is struck with failure.
If we assume that the first and second coins flipped were T and H respectively, then both patterns have 2/3 completion. Something statistically significant happens on the next flip though:
If the third flip is a H, then the THH pattern has been found, and we have to flip at least 3 more times to get a THT pattern.
If the third flip is a T, then the THT pattern has been found, and we have to flip at least 2 more times to get a THH pattern.
The THH pattern has a better chance of coming up because it doesn't overlap itself. A failure for the THT pattern means you have to start all over again, whilst a failure for the THH pattern means you already have the first T and so only need two more flips.
In order to show this in an objective way, I wrote a piece of code: http://pastebin.com/cjkeMnhg
It plays the game 10,000 times (to get a nice sample size), each time flipping 3 coins to start, and then continually flipping until both patterns are found. The average for each pattern is averaged and displayed as the result:
It doesn't stop there though. You can try it with other patterns, and it doesn't matter how big your patterns get, a T followed by a string of H will always come up with fewer tosses on average than the same length alternating pattern:
Statistics is fun! Points go to Paul the Human for the correct answer, and double points go to Zhalentine for posting a solution
Better luck next time everyone else!
The answer is actually that Tails, Heads, Heads comes up with fewer tosses on average. The reasoning is quite easy, but first you need to understand what the question is asking. The question wasn't "Which result requires fewer tosses if you only have 3 tosses" as some people seem to have thought. In that case, the 11 would be correct. If you have only 3 tosses, then all combinations of the 3 tosses (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT) are equally likely of coming up, so THH and THT both have an equal chance (1/8th) of coming up.
However, the question was asking on average, which pattern took the fewest tosses to appear. That means, that if you toss three coins and get HTH, you haven't got either pattern yet and must continue flipping. The same goes for the other 5 patterns that don't equal either THH or THT.
Since we are moving from left to right, we should be able to deduce that both patterns start with TH. This is the common factor between them, and it turns out to be very important. To answer the question, it is necessary to see what happens when each pattern is struck with failure.
If we assume that the first and second coins flipped were T and H respectively, then both patterns have 2/3 completion. Something statistically significant happens on the next flip though:
If the third flip is a H, then the THH pattern has been found, and we have to flip at least 3 more times to get a THT pattern.
If the third flip is a T, then the THT pattern has been found, and we have to flip at least 2 more times to get a THH pattern.
The THH pattern has a better chance of coming up because it doesn't overlap itself. A failure for the THT pattern means you have to start all over again, whilst a failure for the THH pattern means you already have the first T and so only need two more flips.
In order to show this in an objective way, I wrote a piece of code: http://pastebin.com/cjkeMnhg
It plays the game 10,000 times (to get a nice sample size), each time flipping 3 coins to start, and then continually flipping until both patterns are found. The average for each pattern is averaged and displayed as the result:
Program Output Wrote:THH found after an average of 8.0015 (~8) flips.
THT found after an average of 10.0362 (~10) flips.
It doesn't stop there though. You can try it with other patterns, and it doesn't matter how big your patterns get, a T followed by a string of H will always come up with fewer tosses on average than the same length alternating pattern:
Program Output Wrote:THHH found after an average of 16.035 (~16) flips.
THTH found after an average of 20.0729 (~20) flips.
THHHH found after an average of 31.6734 (~32) flips.
THTHT found after an average of 41.9732 (~42) flips.
THHHHH found after an average of 62.9272 (~63) flips.
THTHTH found after an average of 82.5089 (~83) flips.
THHHHHH found after an average of 129.6385 (~130) flips.
THTHTHT found after an average of 169.5909 (~170) flips.
Statistics is fun! Points go to Paul the Human for the correct answer, and double points go to Zhalentine for posting a solution
Better luck next time everyone else!
