Subject 2. Unconditional, Conditional, and Joint Probabilities

The complement of event A is the event that A does not occur. It is expressed as A^c. The probabilities of an event and its complement must have a sum of 1: P(A) + P(A^c) = 1. Note that event A and its complement, A^c, are mutually exclusive (there is no overlapping of their possible outcomes) and exhaustive (these two events include all possible outcomes). For example, event A refers to an increase in interest rates. The probability of event A is 0.7. A^c is the event that interest rates will not increase. P(A^c) = 1 - 0.7 = 0.3.

Probabilities are either unconditional or conditional.

Unconditional probability, also called marginal probability, is simply the probability of an event occurring. It refers to the probability of an event that is not conditioned on the occurrence of another event. For example, what is the probability that a stock earns a return above the risk-free rate? An unconditional probability can be considered as a stand-alone probability. It is expressed as P(A).

A conditional probability is the probability of an event given that another event has occurred. It is denoted as P(A | B) ("the probability of A given B").

For example, what is the probability that the total of two dice will be greater than 8 if the first dice is a 6? This can be computed by considering only outcomes which could occur if the first dice is a 6 and determining the proportion of these outcomes that total more than 8. There are six outcomes for which the first dice is a 6, and of these, there are four that total more than 8 (6,3; 6,4; 6,5; 6,6). The probability of a total greater than 8, given that the first dice is 6, is therefore 4/6 = 2/3. More formally, this probability can be written as: P(total>8 | Dice 1 = 6) = 2/3. In this equation, the expression to the left of the vertical bar represents the event and the expression to the right of the vertical bar represents the condition. Thus, it would be read as "The probability that the total is greater than 8, given that Dice 1 is 6, is 2/3." In more abstract form, P(A|B) is the probability of event A given that event B occurred.

A joint probability is the probability of both events A and B happening together. It is denoted as P(AB) ("the probability of A and B"). For example, Kevin is assessing the probability that both airfare and oil prices increase. Such a probability is a joint probability.

If two events are mutually exclusive, then they cannot occur together, so the joint probability of two mutually exclusive events is 0.

A joint probability function of two random variables, X and Y, gives the probability of joint occurrences of the values of X and Y.

If we know the conditional probability P(A|B) and we want to know the joint probability P(AB), we can use the following multiplication rule for probabilities:

P(AB) = P(A|B) x P(B)

Example 1

If someone draws a card at random from a deck and then, without replacing the first card, draws a second card, what is the probability that both cards will be aces? Event A is that the first card is an ace. Since 4 of the 52 cards are aces, p(A) = 4/52 = 1/13. Given that the first card is an ace, what is the probability that the second card will be an ace as well? Of the 51 remaining cards, 3 are aces. Therefore, p(B|A) = 3/51 = 1/17 and the probability of A and B is: 1/13 x 1/17 = 1/221.

Example 2

The probability of an increase in oil price, P(B), is 0.4. The probability of an increase in airfare given an increase in oil price, P(A|B), is 0.3. The joint probability of an increase in both oil price and airfare, P(AB), is 0.3 x 0.4 = 0.12.

Hint:

• Look out for the words "given that" or "you are told that," which will help you know that the probability is conditional. In the absence of such information, the probability will be unconditional.
• The letter after the | is the event that we know has definitely occurred, whereas the letter before the | is the event whose probability we are trying to calculate.