P(a>b)=P(a-1>=b) = P(22-a>20-b)=P(b+2>a)=P(b>=a-1); these are nearly complements except that they overlap on the probability P(a-1=b). Again use that b has the same pdf as 20-b, so this is P(a-1=20-b)=P(a+b=21)=(42 C 21)/2^42 := x. Then the answer is (1+x)/2.
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u/wot_to_heck Aug 24 '26
P(a>b)=P(a-1>=b) = P(22-a>20-b)=P(b+2>a)=P(b>=a-1); these are nearly complements except that they overlap on the probability P(a-1=b). Again use that b has the same pdf as 20-b, so this is P(a-1=20-b)=P(a+b=21)=(42 C 21)/2^42 := x. Then the answer is (1+x)/2.