Is OP asking for the expected value of the number of steps the man will take in that scenario?
Or is he asking for the probability density function of the random variable for the number of steps taken in the OP's scenario?
>>10033654much less than 10. we will actually converge to the answer pretty quickly just by looking at all possible coin strings of length 4N with progressively larger n and counting how many of them will have heads :tails in ratio 3:1 compared with coin strings that don't satisfy that.
the contributions of long coin strings to the overall expectation value will quickly become tiny.
so for strings of length 4 there are 16 possible strings, 4 of which satisfy the 3:1 ratio, the expected number of steps taken forward is 0.25
for strings of length 8 , there are 256 possible strings , a quarter of which the coins landed in a 3:1 ratio after the 4th coin toss. Of those, again a quarter will land in a 3:1 ratio. so these circumstances contribute 1/16 *2 and 3/16*1
Of the remaining 3/4 of the strings that didn't take a step after the 4th flip we require that after 8 flips they divide in a 3:1 ratio and that both the tails are either the first 4 or the last 4 flips.
there are 256 strings of length 8 of which 8C2 = 28 are in a 3:1 ratio and 12 of those meet our requirements to prevent double-counting.
so the expected value of steps taken after the string is 8 flips long is 1/16*2 +3/16*1 + 12/256*1 = 0.359
so we added less than to the expectation value of just considering strings of length 4.
considering that the contributions of longer stirngs will get smaller and smaller monotonically , we can therefore bound the expectation value as less than the infinite series of geometric series with starting value 0.25 and ratio 0.5 ,
so the expected number of steps taken will be less than 0.5
