The probability of throwing a ``seven'' on two dice can be viewed as the sum of a) the probability of throwing a ``one'' on the first die multiplied by the probability of throwing a ``six'' on the second, plus b) the probability of throwing a ``two'' on the first die multiplied by the probability of throwing a ``five'' on the second, and so forth. Explain the relationship of this observation to the expression for fN~+1(X) in terms of fN~(X−X′) and f1(X′) as given in the Example, and derive the latter expression.