GAME AT MCDONALD’S
4.2 PROBABILITY ESSENTIALS
4.2.3 Conditional Probability and the Multiplication Rule
Probabilities are always assessed relative to the information currently available. As new infor- mation becomes available, probabilities often change. For example, if you read that Kobe Bryant pulled a hamstring muscle, your assessment of the probability that the Lakers will win the NBA Championship would obviously change. A formal way to revise probabilities on the basis of new information is to use conditional probabilities.
Let A and B be any events with probabilities P(A) and P(B). Typically, the probability
P(A) is assessed without knowledge of whether B occurs. However, if you are told that B
has occurred, then the probability of A might change. The new probability of A is called the
conditional probability of A given B. It is denoted by P( ). Note that there is still uncertainty involving the event to the left of the vertical bar in this notation; you do not know whether it will occur. However, there is no uncertainty involving the event to the right of the vertical bar; you know that it has occurred.
AƒB Conditional Probability (4.3) P(AƒB) = P(A and B) P(B) Multiplication Rule (4.4) P(A and B) = P(AƒB)P(B)
The conditional probability formula enables you to calculate P( ) as shown in Equation (4.3). The numerator in this formula is the probability that both A and B occur. This probability must be known to find P( ). However, in some applications P( ) and P(B) are known. Then you can multiply both sides of the conditional probability formula by P(B) to obtain the multiplication rule for P(A and B) in Equation (4.4).
AƒB AƒB
AƒB
The conditional probability formula and the multiplication rule are both valid; in fact, they are equivalent. The one you use depends on which probabilities you know and which you want to calculate, as illustrated in the following example.
4.2 Probability Essentials 161
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4.1 A
SSESSINGU
NCERTAINTY AT THEB
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OMPANYT
he Bender Company supplies contractors with materials for the construction of houses. The company currently has a contract with one of its customers to fill an order by the end of July. However, there is some uncertainty about whether this deadline can be met, due to uncertainty about whether Bender will receive the materials it needs from one of its suppliers by the middle of July. Right now it is July 1. How can the uncertainty in this situation be assessed?Objective To apply several of the essential probability rules in determining the probabil- ity that Bender will meet its end-of-July deadline, given the information the company has at the beginning of July.
Solution
Let A be the event that Bender meets its end-of-July deadline, and let B be the event that Bender receives the materials from its supplier by the middle of July. The probabilities Bender is best able to assess on July 1 are probably P(B) and P( ). At the beginning of July, Bender might estimate that the chances of getting the materials on time from its supplier are 2 out of 3, that is, . Also, thinking ahead, Bender estimates that if it receives the required materials on time, the chances of meeting the end-of-July deadline are 3 out of 4. This is a conditional prob- ability statement, namely, that . Then the multiplication rule implies that
That is, there is a fifty-fifty chance that Bender will get its materials on time and meet its end-of-July deadline.
This uncertain situation is depicted graphically in the form of a probability tree in Figure 4.2. Note that Bender initially faces (at the leftmost branch of the tree diagram) the uncertainty of whether event B or its complement will occur. Regardless of whether event B takes place, Bender must next confront the uncertainty regarding event A. This uncertainty is reflected in the set of two parallel pairs of branches that model whether event A or its complement will occur next. Hence, there are four mutually exclusive outcomes regarding the two uncertain events, as shown on the right-hand side of Figure 4.2. Initially, we are
P(A and B) = P(AƒB)P(B) = (3/4)(2/3) = 0.5 P(AƒB) = 3/4
P(B) = 2/3
AƒB
Figure 4.2 Probability Tree for Example 4.1
P(A and B) = (3/4)(2/3)
P(A and B) = (1/4)(2/3)
P(A and B) = (1/5)(1/3)
interested in the first possible outcome, the joint occurrence of events A and B, found at the top of the probability tree diagram. Another way to compute the probability of both events
B and A occurring is to multiply the probabilities associated with the branches along the
path from the root of the tree (on the left-hand side) to the desired terminal point or outcome of the tree (on the right-hand side). In this case, we multiply the probability of B, corre- sponding to the first branch along the path of interest, by the conditional probability of
A given B, associated with the second branch along the path of interest.
There are several other probabilities of interest in this example. First, let be the complement of B; it is the event that the materials from the supplier do not arrive on time. We know that from the rule of complements. However, we do not yet know the conditional probability , the probability that Bender will meet its end- of-July deadline, given that it does not receive the materials from the supplier on time. In particular, is not equal to . (Can you see why?) Suppose Bender esti- mates that the chances of meeting the end-of-July deadline are 1 out of 5 if the materials do not arrive on time, that is, . Then a second use of the multiplication rule gives
In words, there is only 1 chance out of 15 that the materials will not arrive on time and Bender will meet its end-of-July deadline.
Again, you can use the probability tree for Bender in Figure 4.2 to compute the proba- bility of the joint occurrence of events A and . This outcome is the third (from the top of the diagram) terminal point of the tree. To find the desired probability, multiply the probabilities corresponding to the two branches included in this path from the left-hand side of the tree to the right-hand side. This confirms that the probability of interest is the product of the two rel- evant probabilities, namely 1/5 and 1/3. Simply stated, probability trees can be quite useful in modeling and assessing such uncertain outcomes in real-life situations.
The bottom line for Bender is whether it will meet its end-of-July deadline. After mid- July, this probability is either because by this time, Bender will know whether the materials arrived on time. But on July 1, the relevant probability is
P(A)—there is still uncertainty about whether B or will occur. Fortunately, you can calcu-
late P(A) from the probabilities you already know. The logic is that A consists of the two mutually exclusive events (A and B) and (A and ). That is, if A is to occur, it must occur with
B or with . Therefore, using the addition rule for mutually exclusive events, we obtain
The chances are 17 out of 30 that Bender will meet its end-of-July deadline, given the information it has at the beginning of July. ■
P(A) = P(A and B) + P(A and B) = 1/2 + 1/15 = 17/30 = 0.5667 B B B P(AƒB) = 3/4 or P(AƒB) = 1/5 B P(A and B) = P(AƒB)P(B) = (1/5)(1/3) = 0.0667 P(AƒB) = 1/5 1 - P(AƒB) P(AƒB) P(AƒB) P(B) = 1 - P(B) = 1/3 B
4.2.4 Probabilistic Independence
A concept that is closely tied to conditional probability is probabilistic independence. You just saw how the probability of an event A can depend on whether another event B has occurred. Typically, the probabilities P(A), P( ), and are all different, as in Example 4.1. However, there are situations where all of these probabilities are equal. In this case we say that the events A and B are independent. This does not mean they are mutually exclusive. Rather, probabilistic independence means that knowledge of one event is of no value when assessing the probability of the other.
The main advantage to knowing that two events are independent is that in that case the multiplication rule simplifies to Equation (4.5). This follows by substituting P(A) for
P( ) in the multiplication rule, which is allowed because of independence. In words, the probability that both events occur is the product of their individual probabilities.
AƒB
P(AƒB) AƒB
How can you tell whether events are probabilistically independent? Unfortunately, this issue usually cannot be settled with mathematical arguments; typically, you need empirical data to decide whether independence is reasonable. As a simple example, let A be the event that a family’s first child is male, and let B be the event that its second child is male. Are A and B independent? You could argue that they aren’t independent if you believe, say, that a boy is more likely to be followed by another boy than by a girl. You could argue that they are independent if you believe the chances of the second child being a boy are the same, regardless of the gender of the first child. (Note that neither argument has anything to do with boys and girls being equally likely.)
In any case, the only way to settle the argument is to observe many families with at least two children. If you observe, say, that 55% of all families with first child male also have the second child male, and only 45% of all families with first child female have the second child male, then you can make a good case that A and B are not independent.
It is probably fair to say that most events in the real world are not truly independent. However, because of the simplified multiplication rule for independent events, many mathematical models assume that events are independent; the math is much easier with this assumption. The question is then whether the results from such a model are believable. All we can say in general is that it depends on how unrealistic the independence assumption really is.