The AND rule

The AND rule is applied when you want to know the probability that a set of outcomes occur together (not necessarily at the same time, only that each of the outcomes occurs). It should be obvious that we are talking about outcomes for different events: different outcomes for the same event cannot occur together.

As an example, we might want to know the probability that if you roll 2 dice, the first one is a 1 and the second one is also a 1. Another example would be, if we toss two coins, what is the probability that they are both heads (the first is heads and the second is heads)? A third type of question this rule would apply to is: if two children are born, what is the probability that they are both female?

To answer this type of question, we multiply together the probabilities.

The AND rule:

Pr(A AND B) = Pr(A) x Pr(B)

The most important thing to remember in applying this rule is that the two outcomes must be independent. By this we mean that one outcome does not in some way have an effect on the second outcome (more generally that one event does not influence the other event). With dice it seems obvious that they are independent; it is not clear how rolling the first die would affect the outcome of the second, so the two can be considered independent. In other cases, it might be difficult to determine whether or not two events are independent; the effect of one event on the other can sometimes be extremely difficult to see and there can be many different ways that events affect one another. It is obvious that if two dice are joined by a stick then they will not roll independently, and in such a case you could not apply the AND rule - but things are rarely quite so obvious. In our course on Genetics we will usually be able to assume independence and apply the AND rule but there will be some important exceptions (such as linkage) and we will learn how to deal with these as they arise.

Some common examples of events in genetics that are independent (as far as we know):

• The inheritance of a particular allele from one parent does not affect the inheritance of a particular allele from another parent.


• The genotype (including sex) of one progeny does not affect the genotype of another progeny from the same parents.


• When genes are unlinked the segregation of alleles for the first gene does not affect the segregation of alleles for the second (Independent Assortment!)

More generally, the AND rule can be applied to any number of INDEPENDENT events: