A problem about lottery probability

Fair. If A catches it first, the probability of catching it is1/3; B's second catch was related to A's failure to catch it. If A has caught it, then B will definitely miss it. If A misses, his probability of missing is 2/3. In this case, B has the hope of catching, the probability of catching is 1/2, and the probability of treating B comprehensively is 2/3* 1/2= 1/3, so the probability of catching C is 1/3.