(a) P(at least one red ball) = 1-P(no red ball) = P(both balls blue)
(b) P(second ball red | first ball red)
By the definition of conditional probability, this is equal to
P(first red AND second red)
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P(first red)
I, as well as many students, find the formal definition of conditional probability, in the form of a formula for calculating it, somewhat confusing. For me it is easier to understand this way:
"...given that the first ball is red" means the only cases we are considering are those in which the first ball is red; that means the denominator of the probability fraction, representing the "sample space", is only the probability of getting red with the first ball: P(first red). Then the numerator is the probability that we ALSO get a red on the second ball: P(first red AND second red).
(c) P(second ball red | first ball blue)
This is
P(first blue AND second red)
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P(first blue)