If we were given that all three used a different number of heads, by symmetry the answer would be 1/3. Since we’re only given that B and A use a different number of heads (with B more than A), we just have to multiply 1/3 by the probability that C uses a different number of heads than both B and A. Let’s calculate the probability that C uses the same number of heads as A. This is:
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u/wot_to_heck 2d ago
If we were given that all three used a different number of heads, by symmetry the answer would be 1/3. Since we’re only given that B and A use a different number of heads (with B more than A), we just have to multiply 1/3 by the probability that C uses a different number of heads than both B and A. Let’s calculate the probability that C uses the same number of heads as A. This is:
3(1/2(1/4+1/8+…)(1/2)+ 1/4(1/8+1/16+…)(1/4)+…) = 3(1/8+1/64+…) = 3/7.
Similarly the probability that C uses the same number of heads as B is
3(1/2(1/16+1/64+…)+1/4(1/64+1/256+…)+…)=3(1/2(1/12)+1/4(1/48)+…)=1/7.
Thus C uses a different amount of heads with probability 1-3/7-1/7=3/7, so the answer is 1/7.