2004-04-16
These here probablities - dem are damn hard things, ya'know.
Just sitting around correcting a maths competition for mathematics teachers; when I run across a guy who seriously does this funky trick:
Question: Given that a sequence of cointosses has produce a Head - what's the probability that the next toss is a Head as well?
Answer: The probability for a 1 long streak of heads after this first is 25% (HT)
The probability for a 2-streak is 12.5% (HHT)
The probability for a 3-streak is 6.75% (HHHT)
Summing up this is the sum from 1 to oo of 1/2^n. This sum converges to 1/2. Thus P(H)=0.5
Question: Given that a sequence of cointosses has produce a Head - what's the probability that the next toss is a Head as well?
Answer: The probability for a 1 long streak of heads after this first is 25% (HT)
The probability for a 2-streak is 12.5% (HHT)
The probability for a 3-streak is 6.75% (HHHT)
Summing up this is the sum from 1 to oo of 1/2^n. This sum converges to 1/2. Thus P(H)=0.5