Statistics and Probability Relevant to Genetics

Two basic rules of probability are useful in solving genetics problems: They are the rule of multiplication (or the rule of and) and the rule of addition (or the rule of or).
The Rule of multiplication states that the probability that independent events will occur concurrently is the product of their individual probabilities.
For instance:
Question:
In a Mendelian cross between pea plants that are heterozygous for flower color (Pp), what is the probability that the offspring will be homozygous recessive?
Answer:
•Probability that an egg from the F1 (Pp) will receive a p allele = 1/2.
•Probability that a sperm from the F1 will receive a p allele = 1/2.
•The overall probability that two recessive alleles will fuse, one from the egg and one from the sperm, at the same time, during fertilization is: 1/2 X 1/2 = 1/4.
The Rule of addition is that the probability of an event that can take place in two or more independent ways is the sum of the detached probabilities of the various ways. For instance:
Question:
In a Mendelian cross between pea plants that are heterozygous for flower color (Pp), what is the probability of the offspring being a heterozygote?
Answer:
There are two ways in which a heterozygote may be created: the dominant allele (P) may be in the egg and the recessive allele (p) in the sperm or the dominant allele may be in the sperm and the recessive in the egg.
As a result, the probability that the offspring will be heterozygous is the sum of the probabilities of those two probable ways:
•Probability that the dominant allele will be in the egg with the recessive in the sperm is 1/2 X 1/2 = 1/4.
•Probability that the dominant allele will be in the sperm and the recessive in the egg is 1/2 X 1/2 = 1/4.
•Therefore, the probability that a heterozygous offspring will be produced is 1/4 + 1/4 = 1/2.
The rules of probability can be applied to Mendelian crosses to establish the expected phenotypes and genotypes of offspring.
Important Notes:
•The Product Rule is utilized to determine the outcome of an event with two independent events; the probability of the event is the product of the probabilities of all the individual event.
•The Sum Rule is utilized to determine the outcome of an event with two mutually exclusive events from numerous pathways; the probability of the event is the sum of the probabilities of every individual event.
•The Product Rule of probability is utilized in the determination of the probability of possessing both dominant traits in the F2 progeny; it is the product of the probabilities of possessing the dominant trait for every characteristic.
•The Sum Rule of probability is utilized to determine the probability of possessing one dominant trait in the F2 generation of a dihybrid cross; it is the addition of the probabilities of every individual with that trait.
Probability is a number, between 0 and 1, expressing the exact likelihood of an event taking place.

Probability Basics

The empirical probability of an event is obtained by dividing the number of times the event takes place by the total number of opportunities for the event to occur. Empirical probabilities arrive from observations like that of of Mendel.
An instance of a genetic event is a round seed produced by a pea plant.
Mendel illustrated that the probability of the event “round seed” was guaranteed to occur in the F1 offspring of true-breeding parents, one of which has round seeds and one of which has wrinkled seeds.
When the F1 plants were afterward self-crossed, the probability of any given F2 offspring having round seeds was now three out of four.
In other words, in a large population of F2 offspring chosen at random, 75 percent were anticipated to possess round seeds, while 25 percent were anticipated to have wrinkled seeds.
Making use of large numbers of crosses, Mendel was able to determine probabilities and make use of these to forecast the outcomes of other crosses.
The Product Rule
Mendel illustrated that the pea-plant characteristics he examined were transferred as discrete units from parent to offspring. Mendel as well illustrated that various traits were transferred independently of one another and could be considered in different probability analyses.
For example, performing a cross between a plant with green, wrinkled seeds and a plant with yellow, round seeds gave rise to offspring that had a 3:1 ratio of green: yellow seeds and a 3:1 ratio of round: wrinkled seeds.
The traits of color and texture did not affect every other.
Consider how the product rule is applied to a dihybrid : the probability of having both dominant traits in the F2 progeny is the product of the probabilities of having the dominant trait for every trait.

Role of probability in segregation of alleles and fertilization

In a genetic cross, the probability of the dominant trait being expressed is depends on its frequency. In this case, both parents possessed a dominant and a recessive gene for the trait of flower color. The dominant trait is showcased in 3/4 of the offspring and the recessive trait is expressed in 1/4.
The sum rule can be applied to illustrate the probability of having just one dominant trait in the F2 generation of a dihybrid cross.
To practically make use of probability laws, it is essential to work with a huge sample sizes for the fact that small sample sizes are prone to deviations caused by chance. The large quantities of pea plants that Mendel investigated permitted him to calculate the probabilities of the traits appearing in his F2 generation.
This discovery meant that when parental traits were known, the offspring’s traits could be forecasted correctly even before fertilization.
To make available a scientific context for our probability problems, we will use examples from genetics. Genetics is roughly unique amidst the sciences, in that its basic laws were stated as probability laws.
Thus the probabilities we calculate have an actuality as enduring frequencies, and are not merely subjective.
For instance, the probability a parent of blood-type O has a child of blood-type O is the number of times this event happens among the entire children of all parents of blood type O.
The value of studying genetics is in comprehending how we can forecast the likelihood of inheriting definite traits.
This can help plant and animal breeders in developing varieties that have more improved and desirable qualities. It can as well assist people explain and forecast patterns of inheritance in family lines.
One of the most simple ways to calculate the mathematical probability of inheriting a specific character was discovered by an early 20th century English geneticist known as Reginald Punnett .
His technique makes use of what is currently known as the Punnett square.
This is a simple graphical way of discovering all of the potential combinations of genotypes that can occur in children, given the genotypes of their parents. It as well illustrates to us the odds of every one of the offspring genotypes occurring.
Setting up and using a Punnett square is quite simple once you understand how it works. You start by drawing a grid of perpendicular lines:
Next, you put the genotype of one parent crosswise the top and that of the other parent down the left side. For instance, if parent pea plant genotypes were YY and GG respectively, the arrangement would be:
Take note that only one letter goes in each box for the parents. It does not matter which parent is on the side or the top of the Punnett square.
After that, all you have to do is fill up the boxes by copying the row and column-head letters across or down into the empty squares. This gives us the predicted frequency of all of the potential genotypes amongst the offspring every time reproduction takes place.
In this example, 100% of the offspring will probably be heterozygous (YG). Since the Y (yellow) allele is dominant over the G (green) allele for pea plants, 100% of the YG offspring will have a yellow phenotype, as Mendel discovered in his breeding experiments.
In another example (illustrated below), if the parent plants both possess heterozygous (YG) genotypes, there will be 25% YY, 50% YG, and 25% GG offspring on average.
These percentages are determined based on the fact that everyone of the 4 offspring boxes in a Punnett square is 25% (1 out of 4).
As to phenotypes, 75% will be Y and only 25% will be G. These will be the odds each time a new offspring is conceived by parents with YG genotypes.
An offspring’s genotype is the result of the amalgamation of genes in the sex cells or gametes (sperm and ova) that fuse together in its conception. One sex cell came from every parent.
Sex cells usually have only one copy of the gene for every characteristict ( For example, one copy of the Y or G form of the gene in the example above).
Every one of the two Punnett square boxes in which the parent genes for a trait are placed (across the top or on the left side) in fact represents one of the two possible genotypes for a parent sex cell.
Which of the two parental copies of a gene is inherited depends on which sex cell is inherited–it is a matter of chance.
By placing each of the two copies in its own box has the effect of giving it a 50% chance of being inherited.
Why is it essential t for you to know about Punnett squares is that they can be utilized as predictive tools when considering having children. Let us assume, for example, that both you and your mate are carriers for a specific distasteful genetically inherited disease like cystic fibrosis.
Of course, you are concerned about if your children will be healthy and normal.
For this example, let us use “A” as the dominant normal allele and “a” as the recessive abnormal one that is responsible for cystic fibrosis.
As carriers, you and your mate are both heterozygous (Aa). This disease only afflicts those who are homozygous recessive (aa).
The Punnett square below makes it clear that at every birth, there will be a 25% chance of you having a normal homozygous (AA) child, a 50% chance of a healthy heterozygous (Aa) carrier child like you and your mate, and a 25% chance of a homozygous recessive (aa) child who probably will eventually die from this condition.
If both parents are carriers of the recessive allele for a disorder, all of their children will face the following odds of inheriting it:25% chance of having the recessive disorder 50% chance of being a healthy carrier 25% chance of being healthy and not have the recessive allele at all.
If a carrier (Aa) for a recessive disease mates with someone who has it (aa), the likelihood of their children also inheriting the condition is far greater (as revealed below). On average, half of the children will be heterozygous (Aa) and, therefore, carriers. The remaining half will inherit 2 recessive alleles (aa) and develop the disease
If one parent is a carrier and the other has a recessive disorder, their children will have the following odds of inheriting it: 50% chance of being a healthy carrier 50% chance having the recessive disorder.
It is likely that every one of us is a carrier for a huge number of recessive alleles.