Math Insight

Imagine that we are cross-pollinating two plants, and we are interested in a particular gene that has two variants, or alleles, in these plants, which we'll denote by A and a. Suppose both plants have the genotype Aa, meaning they each have one copy of allele A and one copy of allele a. In this context, an "experiment" is a pollination resulting in a single offspring having one allele from each parent.

In probability terminology, the sample space is the set of all possible outcomes of an experiment. In this case, the outcome will be the genotype of the offspring. List the genotypes that make up the sample space:
. (Separate the genotypes by commas. If a plant has the same allele twice, then we repeat that letter, writing the genotype with two A alleles as AA.)

An event is an outcome or a set of outcomes, and central task is assigning probabilities to different events. For instance, we may wish to know the probability of the event that an offspring also has genotype Aa. Since this event includes just a single outcome, we can refer to it as a simple event. An event can include more than one outcome. The event in which the offspring has at least one copy of the A allele includes two individual outcomes: genotype AA and genotype Aa.

To form a probability model,we assign a probability to each event that indicates how likely each event is to occur. There are some intuitive rules that a probability model must obey. For instance, consider the event in which the outcome is in the sample space. Is it possible for this event not to happen? (Recall the definition of sample space.) yes no

In order to write some of these rules, we need some terminology from set theory.

Let's consider several different events in our cross-pollination example. Say event A is the offspring having at least one copy of allele A, event B is the offspring having at least one copy of allele a, event C is the offspring having genotype aa, and event D is the offspring having genotype AA. What are the specific genotypes in each of these events?

event A:

event B:

event C:

event D:

Observe that A and B have a common genotype,
. This reflects the set theoretic operation of intersection: the intersection of A and B is everything that is in both A and B. In other words, the intersection of A and B is the set of outcomes where both A and B occurred. Symbolically, this is written as A $\cap$ B and read "A intersect B".

Are there any outcomes in A $\cap$ C? yes no
In this case, the sets A and C are called disjoint, and the intersection is called the empty set, or the null set, which is written with the symbol $\emptyset$. Two events are mutually exclusive if their intersection is the null set, because the occurrence of one excludes the possibility of the other.

Suppose we want to describe the event of having two copies of the same allele. There are two ways we can think of this. We can think of it is the combination of events C and D, where any outcome in C or D works, or we can think of it as the opposite of having one of each allele. Let's first look at the combination of events C and D. The union of C and D is the set of everything that is in C or in D. In the language of probability, the union of C and D is the event containing all outcomes in C or in D. Symbolically, we write the union of C and D as C $\cup$ D and read it "C union D". What are the genotypes in C $\cup$ D?

What is A $\cup$ B?

The idea of taking the "opposite" of an event is called the complement. The complement of a set is everything outside of the set. In the case of probability, the complement of an event is the event containing every other outcome in the sample space. We write the complement of A as A$^c$, which is read "A complement". What is A$^c$?
.

What is B$^c$?
What is C$^c$?
.

Let's look briefly at how these operations interact. In the following, write everything in terms of the events X, Y, the null set, and the sample space. These identities apply to all events, and you may find it helpful to think it through with the specific events given above. Write null for the null set and S for the sample space.

(X$^c$)$^c =$

X $\cap$ X$^c =$

X $\cup$ X$^c =$

X $\cup$ (X $\cap$ Y) =

Y $\cap$ (X $\cup$ Y) =

The complement of an intersection can be written as a union, and the complement of a union can be written as an intersection.
(X $\cap$ Y)$^c =$
$\cup$

(X $\cup$ Y)$^c =$
$\cap$

First, as we considered before, the event S, where the outcome is in the sample space, has to happen. Therefore

$1$. $P(S)=1$, where S is the sample space.

Second, probabilities have to be between $0$ and $1$, regardless of the event.

$2$. $0\leq P(A) \leq 1$ for any event A.

Third, if two events are mutually exclusive, the probability of their union is the sum of their probabilities.

$3$. If $A \cap B = \emptyset$, then $P(A \cup B) = P(A) + P(B)$.

From the first and third conditions, we can come up with a useful relationship between the probability of A and the probability of A$^c$. Replace $B$ in the third condition to find $P(A^c)$ in terms of $P(A)$.

$P(A^c) =$

Although the individual outcomes specify the location of each molecule, we aren't interested in knowing which molecule is where. We are only interested in the number of molecules that are still in the cell. Let's define events that reflect the question of interest. Let $n$ be the number of molecules in the cell after $5$ minutes. The events we are interested in are $n=0$, $n=1$, $n=2$, and $n=3$. Are these simple events, or do they contain more than one outcome? No, there are multiple ways to have each event. Yes, each event corresponds to only one possible outcome. Some events are simple events, but not all.

If we know $n=0$, can we tell where each individual molecule is? yes no
. What about if $n=1$? no yes
With $n=0$, every molecule is outside the cell, but with $n=1$, all we know is that one molecule is in the cell, not which one.

In order to determine these probabilities accurately, we need to determine the probabilities of all the simple events.