Wize AP Biology Textbook > Mendelian Genetics
History of Genetics and Laws of Probability
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History of Mendelian Genetics
- Gregor Mendel (1822-1884) was a pioneer in the study of genetics.
- Laws of inheritance derived by hybridization experiments using garden peas.
- Hybridization is the interbreeding between two different varieties of an organism.
- Mendel followed a small number of easily recognizable traits.
- Mendel's strains were true breeding, meaning that the offspring's physical appearance is identical to that of the previous generation. Example: Plants with yellow seeds only produce plants with yellow seeds, plants with purple flowers only produce plants with purple flowers, etc.
Mendel's Objective
His approach was to cross true-breeding strains and cross their progeny to identify any statistical patterns in the frequency of the seven traits. Mendel's approaches differed from other scientist in three important ways:
- The use of true breeding strains rather than complicated and poorly characterized ones.
- A test cross occurs when you breed an organism with one that is homozygous recessive.
- Focusing on just one or a few traits at a time.
- The counting of progeny and looking for statistical patterns.
- P1 Generation: refers to the parental generation.
- F1 Generation: refers to the first offspring generation.
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Basics of Probability for Genetics
Genetics can involve calculating a lot of probabilities. Following are a couple of rules that might help you along the way.
The Product Rule
For interdependent events X and Y, the probability P of them both occurring (X and Y) is the product of the two individual probabilities (PX x PY).
Example: What is the probability that you will get two heads consecutively when flipping a coin?
The probability of getting heads for a regular coin is 1/2 (equal to the probability of getting tails).
Two get two heads in a row, you need the first head (PH,1st = 1/2) and also another head (PH,2nd = 1/2).
Therefore: [ PH,1st x PH,2nd ] = [ 1/2 x 1/2 ] = 1/4
The Sum Rule
For mutually exclusive events X and Y, the probability P that at least one occurs (X or Y) is the sum of the two probabilities (PX + PY) that each individual event will occur.
Example: You're flipping two different coins (coin 1 and coin 2). What is the probability that you will get heads in one coin and tail in the other?
This can be achieved in two ways: coin 1 is heads and coin 2 is tails OR coin 1 is tails, and coin 2 is heads. These two events are mutually exclusive. So:
Probability heads coin 1 = PH1 = 1/2
Probability tails coin 2 = PT2 = 1/2
Similarly for the opposite scenario.
Therefore: [ PH1 x PT2 ] + [ PH2 x PT1 ] = [ 1/2 x 1/2 ] + [ 1/2 x 1/2 ] = 1/2
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Product Rule in Genetics
In a cross between two flowers of the same genotype AaBb, where gene A encodes for one characteristic and gene B for another. Capital letter (A or B) represents the dominant allele, such that AA or Aa will produce the same effect. Calculate the probability of having aabb offspring. (Note: the probability of getting aa is 1/4 and the probability of getting bb is also 1/4).
If you know about Punnett square already you should be able to calculate that Aa x Aa = 1 AA, 2 Aa and 1 aa, so overall aa = 1/4 of the possibilities. This is the exact same for the B gene.
Therefore, in order to get both: Paa x Pbb = 1/4 x 1/4 = 1/16
Probability & the outcome of crosses
Punnett squares are useful for determining the probability of offspring genotypes with monohybrid crosses.
- Example: What is the probability that the offspring will have pink eyes if we crossed two rats, one with black eyes which is hetrozygous dominant, and one with pink eyes which is homozygous recessive.
Black eyed parent: Ee
Pink eyed parent: ee
Probability of pink eyes = 2/4 = 0.5
Once you start looking at multiple genes at a time, determining the probability using Punnett squares is inefficient. Instead, we can use math!
The Product Rule
The probability of two or more independent events occurring at the same time can be calculated by multiplying the individual probabilities of each event.
For example, if you flip one coin your probability of landing on heads is 1/2. If you flip two coins at the same time, the probability of both coins landing on heads is:
(probability of heads on one coin) x (probability of heads on the other coin) =
(1/2) x (1/2) = 1/4
The product rule is also known as the and rule because we use it to determine the probability of event A AND event B happening at the same time. Note that the two events must be independent of each other.
- Example 1: Both parents have the genotype Bb. What is the probability of offspring showing the recessive phenotype?
We can answer this question without using a Punnett square by using the product rule. The only way to produce an offspring with recessive phenotype requires both parents to contribute an gamete with the recessive allele.
Probability of mother producing gamete with "b" allele: 1/2
Probability of father producing gamete with "b" allele: 1/2
Probability of offspring being "bb" = (1/2) x (1/2) = 1/4
This is the exact same result you would get from doing a Punnett square!
- Example 2: Two rabbits were crossed. One parent has white fur (heterozygous) and long ears (homozygous). The other parent has black fur (homozygous) and short ears (heterozygous). What is the probability of offspring with black fur and long ears?
White fur and long ears parent: Ffee
Black fur and short ears parent: ffEe
Probability of black fur: 2/4
Probability of long ears: 2/4
Probability of both black fur and long ears: (2/4) x (2/4) = 4/16 = 1/4
The Sum Rule
The sum rule states that the probability that any of several mutually exclusive events will occur is equal to the sum of the events’ individual probabilities.
For example, if you role a die you have a 1/6 chance of rolling any particular number, but you can only ever roll one number at a time. So you have a 1/6 chance of rolling a 2, and a 1/6 chance of rolling a 3, but the two are mutually exclusive. You cannot roll a 2 and a 3 at the same time. The probability of rolling a 2 OR a 3 is:
(probability of rolling a 2) + (probability of rolling a 3) =
(1/6) + (1/6) = 2/6 or 1/3
The sum rule is also known as the Or Rule because we use it to determine what is the probability of event A OR event B occurring, and the two events are mutually exclusive.
- Example: Both parents have the Bb genotype. What is the probability of the offspring having the dominant phenotype?
Offspring showing the dominant phenotype could be BB OR Bb, so we need to use the sum rule. What are the possible ways we could get offspring with the dominant phenotype?
- Mother contributes B allele and father contributes b allele (offspring are Bb), Or...
- Mother contributes b allele and father contributes B allele (offspring are bB), Or...
- Both mother and father contribute B allele (offspring are BB)
These three events are mutually exclusive, because only one fertilization even can occur at a time. The probability of each of the above events is 1/4 (we can determine this using the product rule, as we did above). To determine the probability of the offspring with the dominant phenotype, we add the three individual probabilities together:
(1/4) + (1/4) + (1/4) = 3/4