How to calculate a genotype with a Rule of Probability. The laws of probability govern Mendelian inheritance
Mendel's laws of segregation and independent assortment reflect the same laws of probability that apply to tossing coins or rolling dice.
The probability scale ranges from 0 (an event with no chance of occurring) to 1 (an event that is certain to occur).
The probability of tossing heads with a normal coin is 1/2.
The probability of rolling a 3 with a six-sided die is 1/6, and the probability of rolling any other number is 1 ? 1/6 = 5/6.
When tossing a coin, the outcome of one toss has no impact on the outcome of the next toss.
Each toss is an independent event, just like the distribution of alleles into gametes.
Like a coin toss, each ovum from a heterozygous parent has a 1/2 chance of carrying the dominant allele and a 1/2 chance of carrying the recessive allele.
The same odds apply to the sperm.
We can use the multiplication rule to determine the chance that two or more independent events will occur together in some specific combination.
Compute the probability of each independent event.
Multiply the individual probabilities to obtain the overall probability of these events occurring together.
The probability that two coins tossed at the same time will land heads up is 1/2 × 1/2 = 1/4.
Similarly, the probability that a heterozygous pea plant (Pp) will self-fertilize to produce a white-flowered offspring (pp) is the chance that a sperm with a white allele will fertilize an ovum with a white allele.
This probability is 1/2 × 1/2 = 1/4.
The rule of multiplication also applies to dihybrid crosses.
For a heterozygous parent (YyRr) the probability of producing a YR gamete is 1/2 × 1/2 = 1/4.
We can use this to predict the probability of a particular F2 genotype without constructing a 16-part Punnett square.
The probability that an F2 plant from heterozygous parents will have a YYRR genotype is 1/16 (1/4 chance for a YR ovum and 1/4 chance for a YR sperm).
The rule of addition also applies to genetic problems.
Under the rule of addition, the probability of an event that can occur two or more different ways is the sum of the separate probabilities of those ways.
For example, there are two ways that F1 gametes can combine to form a heterozygote.
The dominant allele could come from the sperm and the recessive from the ovum (probability = 1/4).
Or the dominant allele could come from the ovum and the recessive from the sperm (probability = 1/4).
The probability of obtaining a heterozygote is 1/4 + 1/4 = 1/2.
We can combine the rules of multiplication and addition to solve complex problems in Mendelian genetics.
Let's determine the probability of an offspring having two recessive phenotypes for at least two of three traits resulting from a trihybrid cross between pea plants that are PpYyRr and Ppyyrr.
There are five possible genotypes that fulfill this condition: ppyyRr, ppYyrr, Ppyyrr, PPyyrr, and ppyyrr.
We can use the rule of multiplication to calculate the probability for each of these genotypes and then use the rule of addition to pool the probabilities for fulfilling the condition of at least two recessive traits.
The probability of producing a ppyyRr offspring:
The probability of producing pp = 1/2 × 1/2 = 1/4.
The probability of producing yy = 1/2 × 1 = 1/2.
The probability of producing Rr = 1/2 × 1 = 1/2.
Therefore, the probability of all three being present (ppyyRr) in one offspring is 1/4 × 1/2 × 1/2 = 1/16.
For ppYyrr: 1/4 × 1/2 × 1/2 = 1/16.
For Ppyyrr: 1/2 × 1/2 × 1/2 = 1/8 or 2/16.
For PPyyrr: 1/4 × 1/2 × 1/2 = 1/16.
For ppyyrr: 1/4 × 1/2 × 1/2 = 1/16.
Therefore, the chance that a given offspring will have at least two recessive traits is 1/16 + 2/16 + 1/16 + 1/16 = 6/16.
Mendel discovered the particulate behavior of genes: a review.
While we cannot predict with certainty the genotype or phenotype of any particular seed from the F2 generation of a dihybrid cross, we can predict the probability that it will have a specific genotype or phenotype.
Mendel's experiments succeeded because he counted so many offspring, was able to discern the statistical nature of inheritance, and had a keen sense of the rules of chance.
Mendel's laws of independent assortment and segregation explain heritable variation in terms of alternative forms of genes that are passed along according to simple rules of probability.
These laws apply not just to garden peas, but to all diploid organisms that reproduce by sexual reproduction.
Mendel's studies of pea inheritance endure not only in genetics, but as a case study of the power of scientific reasoning using the hypothetico-deductive approach.
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