Probability is a method for determining the likelihood of something uncertain occurring. If you flip a coin, you do not know whether it will be heads or tails, but probability can tell you that there is a 1/2 chance of either happening.

If a doctor wants to calculate the probability that a couple’s future offspring will inherit a disease found on a specific genetic locus such as cystic fibrosis, she can also use probabilities.

Consequently, professionals in the medical fields make great use of probabilities as do those in agriculture. Probability helps them with breeding of livestock, with weather predictions for farming and with crop yield predictions for the market.

Probabilities are also essential for actuaries: Their job is to calculate levels of risk for various populations of people for insurance companies so that they know the cost of insuring a 19-year-old male driver in Maine, for example.

A nineteenth-century botanist named Gregor Mendel, and the namesake for Mendelian genetics, used little more than pea plants and mathematics to intuit the existence of genes and the basic mechanism of heredity, which is how traits are passed to offspring.

He observed that his pea plants’ observable traits, or phenotypes, did not always yield the expected ratios of phenotypes in their offspring crops. This led him to conduct crossbreeding experiments, observing the phenotype ratios of each generation of offspring plants.

Mendel realized that traits could sometimes be masked. He had made the discovery of the genotype and had set the field of genetics in motion.

From Mendel’s experiments, he came up with several rules for understanding what must be happening to explain the pattern of trait inheritance in his pea plants. One of them was the law of segregation, which still explains heredity today.

For each trait, there are two alleles, which separate during the gamete formation phase of sexual reproduction. Each sex cell contains only one allele, unlike the rest of the body’s cells.

When one sex cell from each parent fuses to form the cell that will grow into the offspring, it has two versions of each gene, one from each parent. These versions are called alleles. Traits can be masked because there is often at least one allele for each gene that is dominant. When an individual organism has one dominant allele paired with a recessive allele, the individual's phenotype will be that of the dominant trait.

The only way a recessive trait is ever expressed is when an individual has two copies of the recessive gene.

Using probabilities allows scientists to predict the outcome for specific traits, as well as to determine the potential genotypes of offspring in a specific population. Two kinds of probability are especially relevant to the field of genetics:

• Empirical probability
• Theoretical probability

Empirical, or statistical probability, is determined with the use of observed data, such as facts collected during in a study.

If you wanted to know the probability that a high school biology teacher would call on a student whose name started with the letter “J” to answer the first question of the day, you might base it on observations you had made over the past four weeks.

If you had noted the first initial of each student whom the teacher had called on after asking his first question of the class on every school day in the past four weeks, then you would have empirical data with which to calculate the probability that the teacher would first call on a student whose name begins with a J in the next class.

Over the past twenty school days, the hypothetical teacher called on students with the following first initials:

• 1 Q
• 4 Ms
• 2 Cs
• 1 Y
• 2 Rs
• 1 Bs
• 4 Js
• 2 Ds
• 1 H
• 1 As
• 3 Ts

The data shows that the teacher called on students with a first initial J four times out of a possible twenty times. To determine the empirical probability that the teacher will call on a student with a J initial to answer the first question of the next class, you would use the following formula, where A represents the event for which you are calculating the probability:

P(A) = frequency of A/total number of observations

Plugging in the data looks like this:

P(A) = 4/20

Therefore there is a 1 in 5 probability that the biology teacher will first call on a student whose name begins with a J in the next class.

The other type of probability that is important in genetics is theoretical, or classical, probability. This is commonly used to calculate outcomes in situations when each outcome is just as likely to occur as any other. When you roll a die, you have a 1 in 6 chance of rolling a 2, or a 5 or a 3. When you flip a coin, you are equally likely to get heads or tails.

The formula for theoretical probability is different than the formula for empirical probability where A is again the event in question:

P(A) = number of outcomes of in A / total number of outcomes in sample space

To plug in the data for flipping a coin, it might look like this:

P(A) = (getting heads) / (getting heads, getting tails) = 1 / 2

In genetics, theoretical probability can be used to calculate the likelihood that offspring will be a certain sex, or that offspring will inherit a certain trait or disease if all outcomes are equally possible. It can also be used to calculate probabilities of traits in larger populations.

The sum rule shows that the probability of one of two mutually exclusive events, call them A and B, occurring is equal to the the sum of the two individual events’ probabilities. This is depicted mathematically as:

P(A ∪ B) = P(A) + P(B)

The product rule addresses two independent events (meaning that each does not affect the outcome of the other) that happen together, such as considering the probability that your offspring will have dimples and be male.

The probability that the events will occur together can be calculated by multiplying the probabilities of each individual event:

P(A ∪ B) = P(A) × P(B)

If you were to roll a die twice, the formula to calculate the probability that you roll a 4 the first time and a 1 the second time would look like this:

P(A ∪ B) = P(rolling a 4) × P(rolling a 1) = (1/6) × (1/6) = 1/36

In the 1900s, an English geneticist named Reginald Punnett developed a visual technique for calculating the probabilities of offspring inheriting specific traits, called the Punnett square.

It looks like a window pane with four squares. More complex Punnett squares that calculate the probabilities of multiple traits at once will have more lines and more squares.

For example, a monohybrid cross is the calculation of the probability of a single trait appearing in offspring. A dihybrid cross, accordingly, is an examination of the probabilities of offspring inheriting two traits simultaneously, and will require 16 squares instead of four. A trihybrid cross is an examination of three traits, and that Punnett square becomes unwieldy with 64 squares.

Mendel used probability mathematics to calculate the outcomes of each generation of pea plants, but sometimes a visual representation, such as the Punnett square, can be more useful.

A trait is homozygous when both alleles are the same, such as a blue-eyed person with two recessive alleles. A trait is heterozygous when the alleles are not the same. Often, but not always, this means that one is dominant and masks the other.

A Punnett square is particularly useful for creating a visual representation of heterozygous crosses; even when an individual’s phenotype masks the recessive alleles, the genotype reveals itself in Punnett squares.

The Punnett square is most useful for simple genetic calculations, but once you are working with a large number of genes influencing a single trait or looking at overall trends in large populations, probability is a better technique to use than Punnett squares.