# AP: LAB 8: THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

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1 Ms. Foglia Date AP: LAB 8: THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance, but it is also a study of probability. Most eukaryotic organisms are diploid, meaning that each cell contains two copies of every chromosome, so there are two copies of each gene that controls a trait (alleles). In sexual reproduction, these two copies of each chromosome separate, and are randomly sorted into the reproductive cells (gametes). When gametes from two different parents combine in fertilization, new combinations of alleles are created. Thus chance plays a major role in determining which alleles, and therefore which combinations of traits end up in each new individual. The important point is that the inheritance of characteristics is the result of random chance. Therefore, it is important to understand the nature of chance and probability and the resulting implications for the science of genetics. In short, the genes that an individual organism inherits depends on the luck of the draw, and the luck of the draw is dependent on the laws of probability. The Laws of Probability There are three Laws of Probability that are important in genetics and they can be easily demonstrated using simple models like flipping a coin or choosing cards from a deck: The Rule of Independent Events: Past events have no influence on future events. Question: If a coin is tossed 5 times, and each time a head appears, then what is the chance that the next toss will be heads? Answer: 1/2 (1 chance in 2), because coins have 2 sides. The Rule of Multiplication: The chance that two or more independent events will occur together is equal to the product of the probabilities of each individual event. Question: What are the chances of drawing a red nine from a standard deck of cards? Answer: 1/26 (1 chance in 26), because there is 1/2 chance of drawing a red card and 1 chance in 13 of drawing a nine. Therefore, 1/2 x 1/13 = 1/26 or 1 chance in 26 of drawing a red nine. The Rule of Addition: The chance of an event occurring when that event can occur two or more different ways is equal to the sum of the probabilities of each individual event Question: If 2 coins are tossed, what is the chance that the toss will yield 2 unmatched coins (1 head & 1 tail)? Answer: 1/2 (1 chance in 2) because the combination of 2 unmatched coins can come about in 2 ways: Result A (coin #1 heads, coin #2 tails) as well as Result B (coin #1 tails, coin #2 heads). Therefore (1/2 x 1/2) + (1/2 x 1/2) = 1/2, or the chance of Result A plus the chance of Result B. 1 of 10

6 TEAM DATA Toss H/H H/T T/T Toss H/H H/T T/T Toss H/H H/T T/T Total* * The sum of the total of each column must equal 100 tosses. 6 of 10

7 CLASS DATA H/H H/T T/T Totals Obs Exp Total TEAM DATA: CHI SQUARE ANALYSIS H/H H/T A B C D E Obs Exp Obs - Exp (Obs Exp) 2 (Obs Exp) 2 Exp T/T X 2 Total Degrees of Freedom CLASS DATA: CHI SQUARE ANALYSIS H/H H/T A B C D E Obs Exp Obs - Exp (Obs Exp) 2 (Obs Exp) 2 Exp T/T X 2 Total Degrees of Freedom 7 of 10

8 CHI-SQUARE DISTRIBUTION TABLE Degrees Probability (p) value of freedom In general, the probability value of 0.05 is taken as an arbitrary level of significance; if the probability of a fit as poor or poorer than that observed is less than 0.05, the correctness of the hypothesis on which the proposed values are based may be questioned. A probability of more than 0.05 by no means proves that the hypothesis from which we worked is correct but merely tells us that from a statistical standpoint that it could be correct. Furthermore, a probability of less than 0.05 does not prove that a hypothesis is incorrect; it merely suggests that we have reason to doubt the correctness or completeness of one or more of the assumptions on which it is based. 8 of 10

9 QUESTIONS 1. What is the Chi-square test used for? 2. Why is probability important in genetics? 3. Briefly describe how the Chi-square analysis may be used in genetics. 4. Suppose you were to obtain a Chi-square value of 7.82 or greater in your data analysis (with 2 degrees of freedom). What would this indicate? 5. Suppose you were to obtain a Chi-square value of 4.60 or lower in your data analysis (with 2 degrees of freedom). What would this indicate? 6. A heterozygous white-fruited squash plant is crossed with a yellow-fruited plant, yielding 200 seeds. Of these, 110 produce white-fruited plants while only 90 produce yellow-fruited plants. Are these results statistically significant? Explain using Chi-square analysis. What if there were 2000 seeds and 1100 produced white-fruited plants & 900 yellow-fruited? 9 of 10

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