The probabilities are governed by binomial distribution: (p+(1-p))^9=p^9+9p^8(1-p)+(9*8/1*2*)p^7(1-p)^2+...+(1-p)^9. The coefficients are given by the series 1 9 36 84 126 126 84 36 9 1 (Pascal's triangle) and p, the probability of picking a red card is 1/2, so 1-p, the probability of picking a black card, is also 1/2. The first term is the probability of picking 9 reds (or, in this case, 9 blacks); the second term is the probability of picking exactly 2 reds; and so on. We add together probabilities when the question implies OR and we multiply when the question implies AND.
a.
Four or more being red is the same as 1-(3 or fewer being black)=1-(1+9+36)*(1/2)^9=1-46/512=91.02%.
b.
Exactly 2 being red is the third term: 36*(1/2)^9; and exactly 3 is the fourth term 84*(1/2)^9. The probability of EITHER of these is the sum of these probabilities: 120*(1/2)^9=0.2344 or 23.44%.
c.
2 or fewer means 1 or 2, the sum of the first two terms: 10*(1/2)^9=1.95%.
d.
Exactly 5 is 126/512=63/256=24.61%.