Since the alleles (traits) for different characters segregate from one another, probability can be used to mathematically predict the outcomes of genetic crosses involving one or more characters. Although Punnett squares are an effective and straightforward way to examine simple genetic crosses, they are unwieldy and time consuming when the inheritance of multiple characters is examined. A more effective way to examine inheritance is to use two basic laws of probability (The Rule of Multiplication and The Rule of Addition) to determine the probabilities of different outcomes when two individuals are crossed. A little background on probability will help your understanding. Probability scales range from 0 to 1. An event that is certain to occur has a value of 1, whereas an event that is certain not to occur has a value of 0. When you flip a coin, the probability of getting heads is 1/2 or 0.50, and the probability for getting tails is also 1/2 or 0.50. If you are rolling a single die, the probability of rolling a 1 is 1/6 or 0.17, as is the probability of rolling any other number on the die.
If the outcome of a probability event is unaffected by what has happened in previous trials, it is an independent event. When you roll a die, the chance of rolling a 1 is always 1/6, regardless of what you rolled previously. A lottery drawing, on the other hand, is an example of a non-independent event. If balls numbered 1 through 36 are placed in the lottery machine and mixed, the odds of getting any of the numbers is 1/36 on the first number drawn. However, on the second number, the odds decline to 1/35 because the removal of the first ball has affected the odds on the second number. For our purposes, we will be working almost entirely with independent events; however, it is important to understand the distinction.
This figure shows the role of probability when estimating offspring genotypes in Mendelian genetics. The chance of getting a particular offspring requires two independent events to occur: the segregation of alleles into ova (eggs), and the segregation of alleles into sperm. In this example, each of the parents is heterozygous for flower color (Pp); because the two alleles in each individual segregate randomly during meiosis, the chance of getting either a P or pallele in an ova or sperm is 50%. To form an offspring with a particular genotype, the appropriate gametes must come together. The probability of this happening is determined by multiplying the probabilities of the two independent events that must occur to give the desired outcome.
In the example given, the chance of getting a homozygous dominant offspring (PP) would be equal to the chance of getting a P allele in the ova (50%) multiplied by the chance of getting a P allele in the sperm (50%). Much like flipping two coins in the hope of both coming up heads, the probability of getting an offspring of this genotype would be (1/2)(1/2) = 1/4. The same would be true of an offspring with a pp genotype, as the probability of getting two p alleles would also be (1/2) (1/2) = 1/4.
These examples demonstrate the Rule of Multiplication, which allows one to calculate the probability of two or more independent events occurring together in a specific combination. To determine the probability of an event occurring, determine the individual probabilities of each independent event, then multiply the individual probabilities to obtain the probability of these events occurring together. Some non-genetic examples should help to clarify this concept.
(a) What is the probability of getting heads three times in a row when flipping a coin?
The Rule of Multiplication works well when there is only one way to arrive at a particular outcome, but what if the outcome can occur in different ways? For example, what is the probability of getting an offspring with the genotype Pp in the cross shown in this figure? Notice that there are two ways to get this genotype: a P from the ova and a p from the sperm, and vice versa. To calculate this probability, use the Rule of Addition. This rule states that if an event can occur in more than one way, the probability of that event occurring is equal to the sum of the probabilities of each way the event can occur.
As depicted here, the probability of getting an offspring with the genotype Pp is equal to the probability of getting Pp (P from the ova, p from the sperm) plus the probability of getting pP (p from the ova, P from the sperm). Therefore, the probability would be 1/4 + 1/4 = 2/4 = 1/2. This example demonstrates how both the Rule of Multiplication and the Rule of Addition can be used to predict the outcome of genetic crosses. To ensure that you understand the Rule of Addition, here is a non-genetic example.
What is the probability of getting heads, at least once, in two flips of a coin? There are three possible ways to do this: heads on both flips, heads on the first flip, or heads on the second flip.
(a) Use the Rule of Multiplication to calculate the probabilities of each event that satisfies the conditions of the question.
(b) Use the Rule of Addition to calculate the overall probability.
Phenotypes and Genotypes Probability Part 1 VoiceThread Transcript
Phenotypes and Genotypes Probability Part 2 VoiceThread Transcript