Question 1082753
a) P(X=a) = (dCa)((N-d)C(n-a))/(NCn)

in this case: N = 52 (number of total cards)
              n = 5 (number of picked cards)
              d = 26 (number of working cards)

According to the model, it's easier using the complement of the asked probability.

P() = 1 - (P(x=1) + P(x=0) 
    
      1 - (26C1)(26C4)/(52C5) - (26C2)(26C3)/(52C5) = 5,251/9,996 = 52.531%

b) First Not that the E(x) = nd/N = 5/2 ( So the expected number of red cards is between of two and three)

and V(x) = (N-n)(N-d)(nd)/(N^2(N-1)) = 235/204 (So the variation expected is aproximated one between every experiment).

As N = 52 > 30 it's fair to use a Normal aproximation.

y is a Normal with u= 2.5 and sigma square = 235/204

P(y>=2) using standardization P(Z >= (2-2.5)/(sqrt(235/204))

                              P(Z >= -0.46585) using symmetry.

                              0.5 + P(0 <= z <= 0.46585) Extrapolating

                              0.5 + 0.179 = 0.679 = 67.9%

which is different that the exact value, but the error is less than 30%.

If we were using several deck of cards the error would minor.

@natolino_