Two completely unrelated children exhibit allergy symptoms with respective probabilities of 0.5 and 0.3. What is the probability that on a random day that:

If the children are Alan (=0.5) and Barb (=0.3):

  this means:  not Alan = 0.5  and  not Barb  = 0.7:

a) both Alan and Barb:  0.5 and 0.3  =  0.5 x 0.3  =  0.15

b) exactly one:  either Alan and not Barb  or  not Alan and Barb

                                =  0.5 x 0.7  +  0.5 x 0.3  =  0.35 + 0.15  =  0.50

c) neither one:  not Alan  and  not Barb  =  0.5 x 0.7  =  0.35

d) at least one:  both Alan and Barb  or  Alan and not Barb  or  not Alan and Barb

                         =  0.5 x 0.3  +  0.5 x 0.7  +  0.5 x 0.3  =  0.15  +  0.35  +  0.15  =  0.65

With "and", you multiply; with "or", you add.