let's keep working with the example of the coat color of mice where the allele b1 gives rise to a black coat color and the allele b2 gives a white coat color furthermore let's say that black or b1 is dominant our goal right now is to figure out that when you cross or mate to individuals and let's say this is a female the black individual is a female and the white individual is a male when you mate them then what kind of progeny that means what kind of genotype and what kind of phenotype will you get in the progeny these are deployed individuals and therefore the black individual has two of the same type of chromosome that carries the b1 allele whereas the white individual also has two of the same type of chromosome and both of them carry the b2 allele of the gene the first step in mating or crossing these individuals is the production of gametes through meiosis now during meiosis one myocyte will give rise to four gametes two of the myocytes will receive one of the homologous chromosomes whereas the other two gametes will receive the other homologous chromosome however in this case since the individual is homozygous for the b1 allele effectively only one type of gamete will be produced and that gamete will be carrying the b1 allele similarly the the white individual will undergo meiosis and produce only one type of gamete since this individual is also homozygous and this gamete all the gametes will have the b2 allele okay so in carrying out this cross we have completed the first step which is figuring out what are the different gametes different types of gametes each individual will produce the next step is to carry out fertilization and what you're used to as the punnett square is basically carrying out that fertilization on paper so we say that the white individual will produce only one kind of gamete that means all the gametes will be b2 similarly the black individual will produce only one type of gamete carrying the b1 early and then we can visualize fertilization by combining so now we are making a zygote by fertilizing the egg of the the black individual with the sperm of the white individual giving us a progeny who is b1 over b2 now this individual is obviously a heterozygote unlike the parents furthermore all individuals all progeny produced from this cross are going to be heterozygotes third all individuals are going to be black in coat color since black is dominant and these are heterozygotes to review using punnett squares to work out crosses is the same thing as simulating meiosis to determine what kind of gametes and the proportions of different gametes that the parental generation will produce and then carrying out fertilization in the punnett square to determine the genotypes of the progeny next let us work out a particularly simple cross between two homozygous individuals having the same genotype so in our mice example these are both black individuals and they are both homozygous for the b1 allele and this individual will produce only one type of gamete which is b1 similarly this individual also produce only one type of gamete which is b1 and so then when i write down the punnett square i will have all b1 sperm let's say and all b1 x and when you do fertilization you will get all b1 over b1 individuals and this highlights the fact that if you have a population of individuals who are all homozygous for the same allele if you cross any two of them they will always produce this progeny having exactly the same genotype and hence exactly the same phenotype as the parents and therefore this population is called a pure breeding population conversely sometimes in problems you are told that and you meet an individual from a pure breeding population and you can always infer that if an individual is from a pure breeding population that that individual is homozygous for the allele associated with whatever phenotype that individual has so this is a hint in problems when they say pure reading you can always assume that the individual is homozygous now before we take on the next important cross let's talk about a very handy and very important mathematical rule that will help us figure out proportions of progeny of process and this rule is called the product rule so let two random events occur independently so these could be any uh events that are random for example tossing a coin rolling a dice or the random meeting of a sperm and egg during fertilization let the probability of random event one be p1 let the probability of random event 2 be p2 then the probability of event 1 and event 2 occurring together is p1 times p2 please pay attention to the two criterion which allow you to apply the product rule the first thing is that the events should be independent of each other so they should not be able to influence the likelihood of each other's occurrence and the other important criterion for the application of the product rule is that you're computing the probability of event one and event two both events occurring together so let's do an example and let's say the probability of getting a heads on a toss is half so it's a fair coin and i toss the coin twice and the question is what is the probability of two heads so event one is heads on first toss event two is heads on second toss okay and what we would like to know is what is the probability of event one occurring and event two occurring as well which means that you got a heads on the first toss and you got a heads on the second toss and that satisfies the second criterion that it has to be the two events occurring together is the first criterion satisfied yes whether i get heads or tails on the first toss does not influence whether i'll get heads or tails on the second toss and therefore the two events are independent and therefore i can take the probability of getting a heads on the first toss and multiply it by the probability of getting heads on the second toss to obtain the probability of getting two heads in a row using the product rule which would be a quarter so there's a 25 chance that i will get two heads together next let us use the example of the black and white mice to work out a really important type of cross called the monohybrid cross now to remind you b1 over b1 mice are black b2 over b2 mice are white and b1 is dominant in our example now a monohybrid cross is called such because mono means it's a single trait and in this case we are dealing with the single trait of the coat color of mice and it's called a hybrid because it's a cross between two heterozygotes and not only is it a cross between two heterozygotes those two heterozygotes have the exact same genotype over here it is b1 over b2 if this were a plant you could achieve this by self-pollinating and this is also can it can be called selfing now since b1 is dominant it means that both of these mice are black in color let us use a punnett square to determine the genotypes of the progeny and a punnett square as i said before is nothing more than a way of visualizing meiosis and fertilization and let me use green for this head on the right and when this one undergoes meiosis it will produce gametes of two types b1 and b2 and half of these gametes will be b1 and the other half are going to be b2 in other words the probability of getting a b1 gamete is half and the probability of getting a b2 gamete is also half and this is mendel's law of equal segregation and i will use orange for this heterozygote on the left and this this individual also produce half b1 gametes and half b2 gametes so having carried out the meiosis our next step is to carry out fertilization and the b1 gamete when it fertilizes the b1 gamete from the other individual will produce a b1 over b1 zygote what will be the proportion of the b1 over b1 zygote in the progeny and we can use the product rule to compute that now we have two events the first event is obtaining a b1 gamete over here and the second event is obtaining a v1 gamete over here and both of these events are independent because whether we pick a b1 or b2 gamete in the second individual will not influence whether we pick a b1 or b2 gamete in the first individual so that satisfies the first criterion of the product rule that the two events have to be independent and the second criterion is that we are computing the probability of both the events occurring together so event one occurring and event two occurring in this case we are asking whether we pick a b1 gamete over here and a b1 gamete over here so the second criteria of the product rule also applies and then we can compute the probability of this fertilization happening by multiplying the probabilities of the individual events half times half is equal to a quarter similarly when we carry out a fertilization between the p1 gamete and the b2 gaming we will get a heterozygote b1 over b2 and by the product rule this individual will also have a probability of 1 4 or these b1 over b2 individuals will form a quarter of the population of progeny and in this corner we will get a b2 over b1 gaming with a probability of a quarter by the product rule and finally we will get b2 over b2 gametes and once again if we apply the product rule the probability of b2 over b2 or the proportion of b2 over b2 in the population is going to be a quarter having worked out the punnett square for the monohybrid cross next let's determine the ratios of the different genotypes of the progeny as well as the ratios of the different phenotypes that the progeny have but first let me bring over the punnett square that we worked out now it's pretty clear that there are three basic genotypes over here first in blue we have the b1 b1 homozygote let's say in yellow we have the b2b2 homozygote and in pink we have the b1 over b2 heterozygote now note that both the heterozygote in the top right corner as well as the one in the bottom left corner are the same genotypes because it does not matter whether the b1 allele came from the sperm and the b2 came from the egg or it was the other way around that is the b2 allele came from the sperm and the b1 allele came from the egg the genotype if it's a heterozygote is the same and therefore we have a quarter b1 over b1 and we have a quarter plus a quarter which is half heterozygotes and finally we have a quarter b2 over b2 homozygous multiplying these proportions all the way by 4 i get the proportions 1 b 1 over b 1 homozygous is to 2 b1 over b2 heterozygote is to 1 b2 over b2 homozygous and these are the genotypic proportions produced by a monohybrid cross and the next question we can ask is what are the phenotypic ratios and so we can write down the phenotypes and as we had discussed earlier the b1 over b1 individuals are black b2 over b2 individuals are white and since b1 is dominant the heterozygote is going to be black we can fill this in so these individuals have a white coat color that's the phenotype and the heads have a black phenotype since b1 is dominant and finally the b1 over b1 homozygote have a black phenotype because that's uh the v1 allele codes for the black trait the black coat color and this results in a three black is to one white phenotypic ratio a three is to one and it's important to remember that a monohybrid cross will produce a three is to one ratio in favor of the dominant allele and this will come in handy when we are trying to determine which allele is dominant or which allele is recessive when we are given the results of a cross