MODULE NO.22: Probability

Transcription

1 SUBJECT Paper No. and Title Module No. and Title Module Tag PAPER No.13: DNA Forensics MODULE No.22: Probability FSC_P13_M22

2 TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction 3. Laws of Probability 3.1 Product Rule 3.2 Sum Rule 4. Study of Probability in Genetics 5. Examples 6. Summary

3 1. Learning Outcomes After studying this module, you shall be able to Know what is Probability Learn the applications of probability in events Identify the set of conditions where specific rules of probability can be applied Analyze the real time events with respect to probability 2. Introduction Aristotle said, The probable is what usually happens. You can t predict the future, but you can use mathematical probability to determine how likely it is that something will or won t -- happen. Probability is the measure of likelihood that an event will occur or a statement is true. The higher the value of assigned probability with an event more is certainty of the event to occur. Probability values are given between 0 (will not occur) and 1 (will occur). Genetic ratios are most properly expressed as probabilities for example, 3/4 colored: 1/4 white flowers. These values may predict the result of each fertilization event in a way that associates the each resulting zygote having the genetic potential for giving rise to a plant with colored flowers is 3/4 whereas the potential for becoming white flowered plant is 1/4. Random Experiment If an experiment is based on the possibility of having more than one outcome and there is no method or possibility for predicting the outcome prior to occurrence of event or experiment, the event is termed random event. For e.g. in living system, fertilization of gametes is a random event. For such a random experiment a set of all possible outcomes is called sample space which is denoted by s. Event A random experiment may result in several outcomes. A set of all such outcomes of an experiment with a certain characteristic is called an event. An outcome which satisfies a particular characteristic is called favorable outcome and vice-versa. In terms of events, Probability can be expressed as: No of times a particular event occurs Total No of trials during which event could have occurred

4 Events can further be classified into two subtypes: namely independent and mutually exclusive events. Independent Events When the occurrence of two events doesn t not affect the probability of occurrence of one or the other then the two events are said to be independent events. Mutually exclusive Events Mutually exclusive are the two events where the occurrence of outcome of one excludes the occurrence of outcome of other i.e. both cannot occur simultaneously. 3. Laws of Probability 3.1 The Product Law The law states that the probability of two or more events occurring simultaneously is equal to the product of their individual probabilities. Two or more events are independent of one another if the outcome of each one does not affect the outcome of any of the others under consideration. Consider the possible results of an event where you toss a nickel (N) and a penny (P) simultaneously and examine all possible combinations of heads (H) and tails (T) that can occur". The possible outcomes are as follows: P H :N H = (1/2) X (1/2) = 1/4 P T :N H = (1/2) X (1/2) = 1/4 P H :N T = (1/2) X (1/2) = 1/4 P T :N T = (1/2) X (1/2) = 1/4 The probability of obtaining a head or a tail in the toss of either penny or nickel is 1/2 and is unrelated to the outcome of each other. Thus all four possible combinations are predicted to occur with equal probability.

5 3.2 The Sum Law The sum law states that the probability of obtaining any single outcome, where that outcome can be achieved in two or more events, is equal to the Sum of the individual probabilities of all such events. For example, what is the probability of tossing a penny and nickel and obtaining one head and one tail? In such a case, we do not care whether it is the penny or the nickel that comes up heads, provided that the other coin has the alternative outcome. There are two ways in which the desired outcome can be accomplished P H :N T and P T :N H each with a probability of 1/4. Thus, according to the sum law, the overall probability in our example is equal to (1/4) + (1/4) = ½ However, the predictions of all possible outcomes are usually realized only with large sample sizes. This implies that if we predict that 9/16 of the offspring of' a dihybrid cross will express both dominant traits; it is highly unlikely that within a small sample, exactly 9 of every 16 offspring will represent that. Instead, the prediction is that of a large number of offspring, approximately 9/16 representatives will express this phenotype. The deviation from the predicted ratio in smaller sample sizes is attributed to chance. In biological world chance factors account at multiple levels such as fertilization of gametes and so on. Though the impact of deviation strictly attributed to chance is diminished with increase in the sample size. 4. Study of Probability in Genetics 1. Selection of Individual The selected trait that is desired to be studied should be analyzed in individuals on the basis of their phenotypes and genotypes. Following this, individuals with desired trait should be selected for inclusion in study. 2. Transmission of Gene or Chromosome The genetic basis of inheritance of the trait in question should be known and well understood for studying the trait. It is important to know whether the event can occur independently or is mutually exclusive.

6 3. Conditional Probability To calculate the probability of an outcome that is dependent on a specific condition related to that outcome. For example, in the F2 of Mendel's monohybrid cross involving tall and dwarf plants, what is the probability that a tall plant is heterozygous (and not homozygous)? The condition we have set is to consider only tall F2 offspring since we know that all dwarf plants are homozygous. Because the outcome and specific condition are not independent, product law of probability cannot be applied. The likelihood of all such outcomes is referred to as a conditional probability. Conditional probability has multiple applications in genetics. Most significantly, during the genetic counseling it is possible to calculate the probability (P c ) that an unaffected sibling of a brother or sister expressing a recessive disorder is a carrier of the diseasecausing allele (i.e. a heterozygote). Considering that both parents are unaffected (but are carriers for the disorder), the calculation of P c can provide valuable insights on the next child being a carrier or normal or will bear the disorder. To calculate P c, we must consider the probability f, the outcome of interest and that of specific condition (in this case being normal) that includes the outcome. These independent probabilities are P a for Probability of being a heterozygous carrier that receives one recessive allele and one normal allele and P b for probability of not bearing the disorder that is a condition including both homozygous normal and carrier. P a = Probability of inheriting at least one dominant allele = ½ P b = Probability of not bearing disorder (i.e. not having homozygous recessive condition) = ¾

7 In order to calculate the conditional probability P c, We need to divide P a with P b. Therefore, P c = P a / P b P c = ½ / ¾ = 2/3 Thus the probability of having either a carrier or homozygous dominant (normal) child is two third. Additionally probability is only a mathematical model that uses above mentioned principles and predicts theoretical possibilities and there is no certainty that the event will occur. Forked-line method Multihybrid crosses can be analyzed by treating the cross as if it were separate monohybrid crosses and applying the product rule to calculate the probabilities of each outcome. Remember from your previous modules that the Law of Independent Assortment states that the segregation of one gene does not affect the other. Let us consider an example using forked-line method to calculate the probability of each possible phenotype from the cross between Yellow and Round seed bearing plants. YyRr x YyRr

8 Note that the resultant ratio of 9:3:3:1 abides by Mendel s law of independent assortment. Binomial Theorem The binomial formula can be applied only in very specific set of circumstances and determine probabilities of sets of events. Binomial expression can be applied to calculate the probability that a set of events will consist of so many of one type of event and so many of another type of event, and the order of the events does not matter. If the order does matter, the product rule is usually the way applied. Each of the events in the set can have only two possible outcomes. Example: P(a family of 5 has 3 girls and 2 boys) Binomial formula for calculating probability of s type A and t type B events is p s q t Where, n = total number of events s = number of type A events t = number of type B events p = probability of type A event

9 q = probability of type B event Always true: s + t = n, p + q = 1 The factorial function n! (read x factorial) means: n x n-1, x n-2 x... x 1 for example, 5! = 5 x 4 x 3 x 2 x 1 = Examples Let us consider solved examples to understand the applications of probability: Example 1. What is the probability that four children born in a family are all girls? Answer: Let us consider the probability of birth of girl child. Only two possible outcomes are there, hence it is ½. This will apply in the same manner to all four births, each having probability of ½. In order to calculate the probability for 4 children all being girls we need to apply the product rule. Therefore the probability of having all four children being girls is ½ X ½ X ½ X ½ = 1/16 Example 2. One child in every 10,000 births in the U.S. has a genetic disease called phenylketonuria (PKU). a) What is the probability that the next child born in a particular hospital will have PKU? You can assume that the probability of the disease is uniform throughout the US. Answer: P = 1/10,000. b) Suppose a PKU child has just been born in this hospital. What is the probability that the next child born in the same hospital will have PKU? Answer: P = 1/10,000 (Birth of two children is based on the independent set of events that are not correlated with each other. So the fact that one child has already been born with PKU does not change the probability for the next one)

10 c) What is the probability that two children born in a row will have PKU? Answer: P = (1/10,000) (1/10,000) = 1 x 10-8 This question though appears same as that of part (b) but we are asked for the probability that the first child and the next child will both have PKU instead of considering only the second birth. This is why product rule has been applied. The chance of two rare events happening at the same time is exceedingly rare (assuming that they are independent events). It is also noteworthy that in part (b) the first child has already been born; while in part (c), neither of the two events has taken place yet. Example 3. Assuming that the probability of having a girl on any given birth is 1/2, calculate the probability that a family of four will consist of three boys and one girl. Answer: The answer to this question can be easily calculated by writing out all possible birth orders and determining what fraction of the families fit the required family makeup. The possible birth orders: GGGG, GGGB, GGBG, GGBB, GBGG, GBGB, GBBG, GBBB, BGGG, BGGB, BGBG, BGBB, BBGG, BBGB, BBBG, BBBB. Of the 16 possible orders, 4 contain 3 boys and 1 girl, so the probability is 4/16 = 1/4 = The same conclusion can also be reached by using the binomial formula: p s q t (½) 3 (½) 1 = (b) Consider that the true probability of having a boy or girl on any one birth at not quite equal. The actual proportion of girls at birth is about 0.49, rather than Taking this relatively more accurate value, recalculate the probability of having a family of three boys and one girl?

11 Answer: (.51) 3 (.49) 1 6. Summary = 0.26 Probability is the measurement of likelihood whether an event will take place or a statement is true. Probability values are given between 0 (will not occur) and 1 (will occur). If an experiment is based on the possibility of having more than one outcome and there is no method or possibility for predicting the outcome prior to occurrence of event or experiment, the event is termed random event. The Product rule states that the probability of two or more events occurring simultaneously is equal to the product of their individual probabilities The sum rule of probability states that the probability of obtaining any single outcome, where that outcome can be achieved in two or more events, is equal to the sum of the individual probabilities of all such events. If the probability of an out-come that is dependent on a specific condition related to that outcome then, outcome and specific condition are not independent. As a result product law of probability cannot be applied. The likelihood of all such outcomes is referred to as a conditional probability. The binomial expression can be applied only in very specific set of circumstances and determine probabilities of sets of events in order to calculate the probability that a set of events will consist of so many of one type of event and so many of another type of event irrespective of the order of occurrence of events.