The total probability ruleis used to demonstrate how joint probabilities tie in with unconditional probabilities. If we continue with our example from LOS 1.C.gabout interest rates and recession, and assume that the events "I" and "IC" are mutually exclusive and exhaustive, then a recession can only occur with either of these two events. In that case, the sum of these two joint probabilities is the unconditional probability of a recession:
P(R) = P(R given I) * P(I) + P(R given IC) * P(IC)
P(R) = P(RI) + P(RIC)
P(R) = .28 + .06 = .34
l: Define and calculate expected value.
The expected valueis the probability-weighted average of the possible outcomes of the random variable.?
E(X) = ∑xi*P(xi) = x1*P(x

 +x2*P(x2) + … +xn*P(xn)
Here, the "E" denoted expected value. The symbol x1is the first realization of random variable X. The symbol x2is the second realization, etc. In the long run, the realizations should average to the expected value. This is most easily seen using the a priori probabilities associated with a coin toss. On the flip of one coin, we might designate the event "head" as letting the random variable equal one. Alternatively, the event "tail" means the random variable equals zero. A statistician would write:
If head, the X = 1
If tail, then X = 0
For a fair coin where P(head) = P(X = 1) = 0.5 and P(tail) = P(X = 0) = 0.5, the probability weighted average or expected value is:
E(X) = P(X = 0) * 0 + P(X = 1) = 0.5
For the coin flip, X cannot assume a value of 0.5 in any single experiment. Over the long term, however, the average of all the outcomes should be 0.5.
m: Define, calculate, and interpret variance and standard deviation.