Probability, Polygenic, Pedigree Test

As an example, let's use the sum rule to predict the fraction of offspring from an *Aa* x *Aa* cross that will have the dominant phenotype (*AA* or *Aa* genotype). In this cross, there are three events that can lead to a dominant phenotype: *In any one fertilization event, only one of these three possibilities can occur (they are mutually exclusive).* Since this is an *"or" situation* where the events are mutually exclusive, we can apply the sum rule. Using the product rule as we did above, we can find that each individual event has a probability of 1/4. So, the probability of offspring with a dominant phenotype is: (probability of *A* from Mom and *A* from Dad) + (probability of *A* from Mom and *a* from Dad) + (probability of *a/ from Mom and *A* from Dad) =

1/4 + 1/4 + 1/4 = 3/4 *Once again, this is the same result we'd get with a Punnett square.* One out of the four boxes of the Punnett square holds the dominant homozygote, *AA*. Two more boxes represent heterozygotes, one with a maternal *A* and a paternal *a*, the other with the opposite combination. Each box is 1 out of the 4 boxes in the whole Punnett square, and since the boxes don't overlap (they're mutually exclusive), we can add them up (1/4 + 1/4 + 1/4 = 3/4) to get the *probability of offspring with the dominant phenotype.*

If you roll a six-sided die, you have a 1/6 chance of getting any given number, but you can only get one number per roll. You could never get both a one and a six at the same time; *these outcomes are mutually exclusive.* Thus, the chances of getting either a one *or* a six are: (probability of getting a 1) + (probability of getting a 6) =

1/6 + 1/6 = 1/3

If you roll a six-sided die once, you have a 1/6 chance of getting a six. If you roll two dice at once, your chance of getting two sixes is: (probability of a six on die 1) x (probability of a six on die 2) = ________

1/6 x 1/6 = *1/36*

What phenotypic ratio is seen in a dihybrid cross? *(Hint both parents are always heterozygous)*

9:3:3:1

Law of Segregation

A sperm/egg carries only *one allele* for each inherited character because *allele pairs (homologous chromosomes)* separate from each other during *gamete formation*

What are ways of determining whether a trait is dominant or recessive (in a pedigree)?

If the trait is dominant, one of the parents must have the trait and the trait must then be passed onto the next generation. If the trait is recessive, neither parent is required to have the trait since they can be heterozygous.

If the law of independent assortment was not present what would be different about the results of a dihybrid cross?

If two genes didn't follow independent assortment or if independent assortment was not present then the two different genes might have been inherited together. Ex: Seed color and seed shape might have been inherited as a pair. yellow and round alleles might have stayed together and the green and wrinkles alleles might have stayed together. P - YR YR x yr yr (each allele is passed on as a unit so YR, yr, etc) F - YR yr -----> self-fertilization F2 - YR yr x YR yr Results in only two phenotypic classes - 3:1 (yellow round seed: green wrinkled) If the seed color and seed shape genes were always inherited as a unit, or *completely linked*, a dihybrid cross should produce two types of offspring, *yellow/round* and *green/wrinkled*, in a *3:1 ratio.*

When during meiosis are the laws of segregation and independent assortment apparent?

Independent assortment - Metaphase I Laws of Segregation - Anaphase I

What is the difference between the law of segregation and the law of independent assortment?

Law of Segregation - Two alleles of a single trait will separate randomly meaning there's 50% chance either allele will end up in either gamete (monohybrid cross; 3:1) Law of Independent Assortment - The allele of one gene separates independently of an allele of another gene (dihybrid cross; 9:3:3:1). The allele a gamete receives for one gene does not influence the allele received for another gene.

What are the symbols for male and female (in a pedigree)?

Male - Square Female - Circle

What is the difference between a monohybrid cross and a dihybrid cross?

Monohybrid cross - Both parents are heterozygous for a single trait (Aa x Aa, Bb x Bb) Dihybrid cross - cross of two different traits (ex AaBb x AaBb, CcDd x CcDd)

What is different between polygenic traits and monogenic traits?

Polygenic - When multiple genes, not necessarily on the same chromosome, affect inheritance/phenotypic characteristic. Monogenic - One trait controlled by a single gene or allele.

a) What is the probability of getting heads three times in a row when flipping a coin? b) What is the probability of rolling a six on two consecutive rolls of a die?

Probability of getting heads on one flip = 1/2 Probability of getting heads on three consecutive flips = 1/2 x 1/2 x 1/2 = *1/8* b) Probability of getting a six on one roll = 1/6 Probability of getting a six on two consecutive rolls = 1/6 x 1/6 = 1/36

What does the distribution of polygenic phenotypes typically look like? Why?

Resembles a *bell curve.* Due to *continuous variation*. There are multiple genes that play a role in the phenotype, and each gene could have multiple alleles. Polygenic traits produce many different phenotypes due to a gene having two or more alleles assigned so a bell curve is effective to graph many phenotypes and their distribution.

How do you determine the probability of producing a particular gamete?

Rule of Multiplication Requires two independent events to occur: the segregation of alleles into egg of the female, and the segregation of alleles into sperm in the male. For example, say 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 *p* allele in an egg or sperm is *50%*

What is the difference between a filled in shape and a blank shape (in a pedigree)?

Shaded - Affected/showing symptoms Blank - Unaffected

If you cannot directly determine an individual's genotype (in a pedigree) what should you do?

That sometimes means the individual is either homozygous or heterozygous for a trait. Normally, we can use the relationships between an individual and their parents, sibling, and offspring to determine genotypes. Basically do a Punnett square of the individual's parents, Ex: *Aa* x *Aa* (*A* is dominant and individual expressed the dominant trait) 1/4 *AA* 1/2 *Aa* 1/4 *aa* Can't determine whether the individual is heterozygous or homozygous dominant. So you say the individual is *1/4 AA and 1/2 Aa*

What is the difference between the rule of multiplication and the rule of addition?

The Multiplication rule is for calculating the probability of two (or more) independent events occurring together. The Addition rule is for calculating the probability of several mutually exclusive events (if only one or the other event can occur).

What is the chance of getting a particular offspring for one trait (With Rule of Multiplication)?

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. For example, say each of the parents is heterozygous for flower color (*Pp*); The chance of getting a *homozygous dominant* offspring (*PP*) would be *equal to the chance* of getting a *P* allele in the egg (*50%*) multiplied by getting a *P* allele in the sperm (*50%*). *Think of flipping two coins in the hope of both coming up heads* The probability of getting an offspring of this genotype (*PP*) would be 1/2 x 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 x 1/2 = *1/4*

Law of Independent Assortment

The inheritance of *one characteristic (gene)* has no effect on the inheritance of another (gene) because alleles on non-homologous chromosomes assort independently during gamete formation

We can use the product rule to predict frequencies of fertilization events. For instance, consider a cross between *two heterozygous (Aa)* individuals. What are the odds of getting an *aa* individual in the next generation?

The only way to get an *aa* individual is if the *mother* contributes an *a* gamete and the *father* contributes an *a* gamete. Each parent has a 1/2 chance of making an *a* gamete. Thus, the chance of an *aa* offspring is: (probability of mother contributing *a*) x (probability of father contributing *a*) = 1/2 x 1/2 = 1/4

Rule of Multiplication

The probability of *two (or more) independent events* *occurring together* can be calculated by multiplying the individual probabilities of the events. In general, you can think of the product rule as the *"and" rule*: *if both event X and event Y*must happen in order for a certain outcome to occur, and *if X and Y are independent* of each other (don't affect each other's likelihood), then *you can use the product rule* to calculate the probability of the outcome by *multiplying the probabilities of X and Y.*

Rule of Addition

The probability that any of *several mutually exclusive events* will occur is *equal* to the *sum of the events' individual probabilities.* You can think of the sum rule as the *"or" rule*: if an outcome requires that *either event X or event Y* occur, and if X and Y are *mutually exclusive (if only one or the other can occur in a given case),* then the *probability of the outcome* can be calculated by *adding the probabilities of X and Y.*

Non-independent event

Two or more events in which the outcome of one event affects the outcomes of the other event(s). Ex: a lottery drawing. 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.

Independent Event

Two or more events in which the outcome of one event does not affect the outcome of the other event(s). Ex: the chance of rolling a 1 on a six-sided die is always 1/6.

What is the chance of getting a particular offspring for one trait (With Rule of Addition)?

Use if the outcome can occur in different ways rather than when there is only one way to arrive at a particular outcome (that's for the product rule). For example, say each of the parents is heterozygous for flower color (*Pp*); What is the probability of getting an offspring with the genotype *Pp*? There are two ways to get this genotype: a *P* from the egg and a *p* from the sperm, and vice versa. Basically a *Pp* x *Pp* cross and you're trying to find the probability of an offspring with *Pp* The probability of getting an offspring with the genotype *Pp* is equal to the probability of getting *Pp* (*P* from the egg, *p* from the sperm) plus the probability of getting *pP* (*p* from the egg, *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.*

What is the difference between X-linked, Dominant/Recessive, and Incomplete Dominance?

X-Linked - Are commonly recessive. Occur more frequently in males than females. If X-linked recessive, all offspring of a carrier female have a 25% chance of inheriting the mutation. All female children of an affected father will be carriers. Dominant/Recessive - Dominant allele masks the presence of the recessive allele. One copy of the dominant allele needed to express the trait. Incomplete Dominance - Neither allele for a gene dominates. The heterozygous phenotype is a blend of the two homozygous parents. Ex: true-breeding red and white flowers have pink flowers.

What are ways of determining whether a trait is X-linked or autosomal (in a pedigree)?

X-Linked - For X-linked recessive traits, males are much more affected than females (1 X vs 2 X) Autosomal - Both males and females are equally likely to be affected (usually in equal proportions)

What is the difference between X-linked notation and autosomal (regular) genotype notation?

X-Linked - Written as superscripts on the X chromosome. No alleles are written on the Y chromosome. Autosomal - The trait (gene) is represented by one letter. Dominant allele - capital letter Recessive allele - lowercase letter

How do you calculate the probability of an offspring inheriting a particular genotype from its parents for more than one trait at a time? (Dihybrid cross ex: BbCc x BbCc) Example to answer the first question: Imagine that we breed two dogs with the genotype *BbCc*, where dominant allele *B* specifies *black coat color* (versus *b*, *yellow coat color*) and *dominant allele C* specifies *straight fur* (versus *c*, *curly fur*). Assuming that the two genes *assort independently* and are *not sex-linked*, how can we predict the number of *BbCc* puppies among the offspring?

You can do a dihybrid cross but for a cross involving two genes, you can also use four-square Punnett squares and the Product rule. Break the overall question down into two smaller questions, each relating to a different genetic event: a. What's the probability of getting a *Bb* genotype? b, What's the probability of getting a *Cc* genotype? In order for a puppy to have a *BbCc* genotype, *both of these events* must take place: the puppy must receive *Bb* alleles, *and* it must receive *Cc* alleles. That "and" right there? That tells us that we'll have to use the product rule which means the two events are independent. (Also because the genes assort independently) So, once we calculate the probability of each genetic event, we can multiply these probabilities using the product rule to get the probability of the genotype of interest (*BbCc*). Using a Punnett square, you can see that the probability of the *Bb* genotype is 1/2. (Alternatively, we could have calculated the probability of *Bb* using the product rule for gamete contributions from the two parents and the sum rule for the two gamete combinations that give Bb.) Using a similar Punnett square for the parents' fur texture alleles, the probability of getting an *Cc* genotype is also 1/2. To get the overall probability of the *BbCc* genotype, we can simply multiply the two probabilities, giving an overall probability of 1/4.

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. Probability of getting heads on first flip, heads on second flip = 1/2 x 1/2 = 1/4 Probability of getting head on first flip, tails on second flip = 1/2 x 1/2 = 1/4 Probability of getting tails of first flip, heads on second flip = 1/2 x 1/2 = 1/4 b) Use the Rule of Addition to calculate the overall probability. *The probability of getting heads, at least once, in two flips of a coin = (heads on the first flip, tails second) + (tails first, heads second) + (heads both times) =* 1/4 + 1/4 + 1/4 = 3/4 The probability of getting heads, at least once, in two flips of a coin = 3/4