Project 4

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Northeastern University *

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1151

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Biology

Date

Feb 20, 2024

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pdf

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Project Four: Fun with Numbers This week you'll learn about some uses of basic probability in genetics, such as how to estimate the probability of specific outcomes in genetic crosses, and a method for determining whether observed data fit what is predicted with respect to some hypothesis – a concept known as "goodness of fit". Read the text sections of this project. View this ~10-minute video on probabilities to get started, keeping in mind that this field of mathematics has its origins in people trying to game the odds in betting games! Watch this ~10-minute video on testing hypotheses using a chi-squared test. Pre-class prep : Multiplication rule: The probabilities of independent events are multiplied. Example: when rolling a single six-sided die, the probability of rolling a four is 1/6 since there are six possibilities in total. What is the probability of rolling a four in the first roll (1/6) AND a four in the second roll (1/6)? Since what you roll the second time is in no way influenced by what you rolled the first time, these two events are independent, and to find the probability that they both happen, you multiply the probability of each: 1/6 x 1/6 Addition rule: The probabilities of mutually exclusive events must sum to one. Example: when rolling a single die, you will either roll a 1 OR 2 OR 3 OR 4 OR 5 OR 6. There are no other options, and each is mutually exclusive, so the probabilities must add up to one. 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1. What is the chance that you roll a three OR a four on one roll? 1/6 + 1/6 Each question is worth one point unless otherwise specified. 33 points total. I. Casey at the casino (7 pts) From a deck of 52 cards, you select one card at random. Give the probability for each occurrence. 1. Casey selects a 9: 1/13 2. Casey doesn’t select a 9: 25/26 3. Casey selects a 9 or 4: 2/13 4. Given that Casey has selected a face card (jack, queen, or king), Casey selects the Queen of hearts: 1/12 5. Given that Casey has selected a black card, Casey selects a spade: 1/4

6. Casey selects a card from the deck, then the dealer reinserts the card into the deck and shuffles, and Casey picks a card again. Both cards are kings: 1/169 (4/52 * 4/52) 7. Casey selects a card from the deck but the dealer does not reinsert it before Casey picks a card again. Both cards are kings: 1/221 (4/52 * 3/51) II. Probability and inheritance (6 pts) Diploid organisms carry two copies of every gene--one on the maternal chromosome, one on the paternal chromosome. Both copies (alleles) may be the same, or different. Consider a plant with a gene for flower color, where one allele produces purple flowers and the other allele produces white flowers. When the plant produces gametes, each gamete receives one or the other allele, randomly, like a coin toss. Many interesting findings concerning patterns of inheritance of genetic information have been deduced by observing the outcomes of crosses between organisms with different alleles for easy-to-observe traits, such as flower color in plants, or wing shape and eye color in fruit flies. Laws of probability allow researchers to predict the outcomes of crosses. For example, Mendel determined long ago that a gamete has an equal chance of receiving the paternal or maternal allele for a gene. This finding is known as the Law of Segregation. If a plant has one allele for purple flowers (let's denote it as P) and one allele for white flowers (p), each gamete has an equal probability of receiving P or p, just as a coin toss has an equal chance (if the coin is fair) of producing heads or tails. Remember that one of the alleles is from the paternal copy of the genome, and the other is from the maternal copy of the genome. A plant with alleles PP will produce purple flowers, and a plant with alleles pp will produce white flowers. Of course, one trait is usually dominant over the other, meaning, when an organism is heterozygous at that locus (having one of each allele) only the dominant trait is expressed. In our plant example, purple is dominant and white is "recessive", so a plant with the genotype Pp produces purple flowers, just like a plant with PP genotype. Only a plant with the genotype pp, meaning, homozygous (having two identical alleles) for the recessive trait, produces white flowers. Punnett squares illustrate two applications of probability--the sum rule and the product rule. The sum rule applies to mutually exclusive events, and the product rule to independent events. Consider the likelihood of whether a gamete receives a P or p allele when both parents are heterozygotic for that gene. Each time a gamete is formed, it has an equal chance or receiving P or p, regardless of what the previous gamete received. Similarly, the zygote that will form from the joining of two gametes has an equal chance of receiving either one of the two alleles from one parent, and either one of the two alleles from the other. The inner squares show the probability for every possible zygote that will result from a union of parental gametes. Note that the probabilities add up to 1. The probability that a zygote will have a specific

combination of maternal/paternal alleles is 1/4, but the probability that it will be some combination from among the set of four is 100%, or 1. Diploid organisms carry two copies of every gene--one on the maternal chromosome, one on the paternal chromosome. Both copies (alleles) may be the same, or different. Consider a plant with a gene for flower color, where one allele produces purple flowers and the other allele produces white flowers. When the plant produces gametes, each gamete receives one or the other allele, randomly, like a coin toss. Many interesting findings concerning patterns of inheritance of genetic information have been deduced by observing the outcomes of crosses between organisms with different alleles for easy-to-observe traits, such as flower color in plants, or wing shape and eye color in fruit flies. 8. What is the chance that a cross between two heterozygotic parents results in heterozygotic offspring? 50 percent 9. What fraction of offspring of two heterozygotes is expected to be homozygous recessive (carry two recessive alleles)? 1/4 10. What is the probability that a cross between parents with genotypes PpWWrr and PpWwRR produces an offspring with genotype PPWWRr? (You may need to refer back to the video on probabilities). 1/8 11. A couple plans to have five children. How many possible combinations of males and females are possible? 32

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