HANDBOOK for PROBABILITY CAlCUlatIons Many problems in diploid genetics rely on basic concepts of probability. This is because each individual inherits at random only one of two possible copies of a gene from each parent. Thus, breeding experi ments or inheritance in human pedigrees have probabilistic rather than absolute outcomes. Everyone has an intuitive sense of probability but what we need is a precise definition that will allow probabilities to be manipulated quantitatively Probabilities are usually defined in terms of possible outcomes of a trial. A trial could be the toss of a coin, the roll of a die, or two parents having a child. If we define a specific event a, p(a) or the probabil ity of a, can be defined as follows: after a very large number of trials, p(a) is simply the fraction of trials that give outcome a. In principle, we could determine pla by actually performing a large number of trials and directly measuring the fraction of trials that produce event a. This is sometimes called the Monte Carlo method " named after a famous European casino and works well for computer simulations of complicated phenomena. However, in many cases there is a much simpler way to calculate probabili ties. To directly calculate classical probabilities one must know enough about a process to break down the possible outcomes of a trial into some number of equally probable events. In these cases the prob- ability of event a is (a n where na is the number of outcomes that satisfy the criteria for a and n is the total number of equally probable outcomes. Note that since n includes all possible outcomes, na sn and o sp(a)sl Example: A couple has two children, what is the probability that they are both girls? Assuming that the chances of having a boy or a girl are equal, there are 4 equally probable ways of having two children (boy, boy; girl, boy; boy, girl; girl, girl) and the probability of two girls is 1/4 or 0.25 For classical probability problems you will always be able to arrive at the correct answer by writing out all of the possible outcomes of a trial and counting the fraction of outcomes that satisfy the criteria for a given event. Often, enumerating all of the outcomes for a trial is time-consuming and error-prone. It is usually faster and easier to break a problem down into simple parts and then to combine the probabilities for the individual parts. The following are useful ways that probabilities can be combined to speed obability calculation PRODUCTRULE p(a and b)=p(a) p(b) if a and b are independent Two events are considered independent if they do not influence one another. The criterion of indepen dence is very important-application of the product rule for events that are not independent will give an incorrect answerHANDBOOK for PROBABILITY CALCULATIONS Many problems in diploid genetics rely on basic concepts of probability. This is because each individual inherits at random only one of two possible copies of a gene from each parent. Thus, breeding experi￾ments or inheritance in human pedigrees have probabilistic rather than absolute outcomes. Everyone has an intuitive sense of probability but what we need is a precise definition that will allow probabilities to be manipulated quantitatively. Probabilities are usually defined in terms of possible outcomes of a trial. A trial could be the toss of a coin, the roll of a die, or two parents having a child. If we define a specific event a, p(a) or the probabil￾ity of a, can be defined as follows: after a very large number of trials, p(a) is simply the fraction of trials that give outcome a. In principle, we could determine p(a) by actually performing a large number of trials and directly measuring the fraction of trials that produce event a. This is sometimes called the “Monte Carlo method” named after a famous European casino and works well for computer simulations of complicated phenomena. However, in many cases there is a much simpler way to calculate probabili￾ties. To directly calculate classical probabilities one must know enough about a process to break down the possible outcomes of a trial into some number of equally probable events. In these cases the prob￾ability of event a is: p(a)= na N where na is the number of outcomes that satisfy the criteria for a and N is the total number of equally probable outcomes. Note that since N includes all possible outcomes, na ≤ N and 0 ≤ p(a) ≤ 1. Example: A couple has two children, what is the probability that they are both girls? Assuming that the chances of having a boy or a girl are equal, there are 4 equally probable ways of having two children (boy, boy; girl, boy; boy, girl; girl, girl) and the probability of two girls is 1/4 or 0.25. For classical probability problems you will always be able to arrive at the correct answer by writing out all of the possible outcomes of a trial and counting the fraction of outcomes that satisfy the criteria for a given event. Often, enumerating all of the outcomes for a trial is time-consuming and error-prone. It is usually faster and easier to break a problem down into simple parts and then to combine the probabilities for the individual parts. The following are useful ways that probabilities can be combined to speed probability calculations. PRODUCT RULE p(a and b) = p(a) x p(b) if a and b are independent. Two events are considered independent if they do not influence one another. The criterion of indepen￾dence is very important — application of the product rule for events that are not independent will give an incorrect answer