What are the Odds? Page 1 of 7. Tutorial 10f: Probability


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1 Page of 7 Module #: Tutorial 0f: Probability What are the Odds? Wouldn t you like to know the probability of winning a state lottery? You and everyone else! Maybe you have wondered what the odds are of your own children inheriting your green eyes or your dimples. You may even be concerned about whether you have a risk of developing a disease that runs in your family, or whether you could pass that risk on to your children. Nobel laureate Paul Berg recently addressed a cancer symposium and said that many human diseases are genetic in origin. In other words, your genes may affect whether you may develop a psychiatric disorder, cystic fibrosis, some cancers as well as the physical traits such as color of eyes and height. The P Team (Probability Team) is setting out to explore probability and genetics. Travel with them and you will: Gain an understanding of the laws of probability Discover the basics about genetics Be able to determine many probabilities for yourself The P Team discovered the Fundamental Counting Principle: If a first event can happen x ways, and a second event can happen y ways, then the two events can occur in x y ways. If there is a third or fourth event, or even more, then you would continue to multiply by each new event: x y z k, et What are the Odds? I: Genetics The P Team poses some practice questions on the Fundamental Counting Principle for you: ) Genetics have a lot to do with eye and hair color. If you could choose both, how many combinations are there when given a choice of green, blue and brown eyes and red, brown, blond and black hair? a = 6 combinations b. 3 3 = 9 combinations 3 + = 7 combinations 3 = combinations ) Finish filling in the table: Combinations Red hair Brown hair Blond hair Black hair Green eyes Green, Red Green, Brown Green, Blond Green, Black Blue eyes Blue, Red,,, Brown eyes Brown, Red,,, 3) Many humaninherited traits are influenced by genes. Some of these are simple dominant and recessive traits. Dominant Recessive Event Has dimples Lacks dimples Event Can roll tongue Cannot roll tongue Event 3 Unattached ear lobes Attached ear lobes Event Long eyelashes Short eyelashes Event Curly hair Straight hair Those who have dimples express the dominant trait. Those who lack dimples express the recessive trait. One can either have it or not in each event. In other words, they either have the dominant trait or the recessive trait. If the fundamental counting principle applies to the characteristics of a person, then how many different combinations are there with these five human inherited traits? a. combinations b. 0 combinations combinations 3 combinations Copyright 00. All rights reserve 97
2 Page of 7 Module #: Tutorial 0f: Probability The Probability Team needed some organization and decided to designate two leadership roles amongst themselves. There were five members to begin with and they wanted to figure out how many different possible pairs of leaders they could choose. What are the Odds? II: Genetics cont... If the team decided to pick a president and a head researcher, then there are two methods used to organize the arrangements. The first method uses Permutations. Permutations: An arrangement of items where each different order counts as a different arrangement. President Head Researcher The first tree shown has President and Head Researcher. The second tree shown has President and Head Researcher. These are different permutations because the order matters. If you observe all the trees, there are 0 permutations. Answer the team s questions below involving permutations: ) Selective breeding has been around for years. Each year farmers choose only the seeds from the best cornstalks to plant the next year. If six farmers are competing for st and nd prize for the tallest cornstalks, how many permutations for the two prize winners are there? a. 6 b Another way to organize these permutations is by looking at a chart:,,,, 0 permutations,,,,,,,,,,,,,,,, You can use multiplication to find all the possible permutations for a set of items: President Head Researcher possible people people remaining to choose = 0 permutations Suppose all team members wished to have their own special title. How many permutations would be possible? 3 = 0 permutations. Sometimes disorders are inherite These are traits that many people consider abnormal. ) Six people are waiting in line at a doctor s office where they test for Redgreen colorblindness (an inherited trait where a person is unable to tell the difference between red and green.) How many permutations are possible to arrange all six people in the line? a. 6 b Copyright 00. All rights reserve 9
3 Page 3 of 7 Module #: Tutorial 0f: Probability The Probability Team could also choose two leaders without assigning them the roles of President and Head Researcher. They analyze the number of combinations of leaders from their group of five. What are the Odds? III Genetics cont... If the team decided not to pick a president and a head researcher, but to just have two people in equal leadership, then the Combination method would apply. Combination: An arrangement of items where the different order of items does not count as a different arrangement. Leader Leader The first tree shown has leaders and groupe The second tree shown does not group with. This would be a repeat of that combination. If you observe all the trees, there are 0 combinations. (No repeating combinations may occur) The P Team has some questions for you to answer involving combinations: 6) A crime lab is testing DNA (nucleic acid carrying genetic information) from possible suspects. They know there are two people who committed the crime. How many different combinations might there be? a. b. 6 6 Another way to organize these combinations is by looking at a chart. The repeated combinations are shaded and cannot be counte,,,,,,,,,,,,,,,,,,,, You can compute to find all the possible combinations for a set of items: Step : Leader Leader people choice people choice = 0 permutations Step : Any combination can be arranged ways. & then & Step 3: Divide the permutations by how many arrangements of any combinations: 0 = 0 Therefore, there are 0 combinations. units called genes. These genes contain the information about one, small aspect of the organism. The DNA then is organized into chromosomes, which is a long stretch of DNA. Many organisms (diploids) have two parents, and so receive two sets of DNA. For example, humans have 6 chromosomes, receiving 3 chromosomes from each parent. 7) If a geneticist wanted to only look at of the 6 chromosomes from a human, how many different combinations are there? a. 6 b DNA specifies pigmentation, size and shape of organs. This DNA is made of many Copyright 00. All rights reserve 99
4 Page of 7 Module #: Tutorial 0f: Probability The team is now interested in finding the probability of an event happening. At this point they know the Fundamental Counting Principle and know how to compute permutations and combinations. They are very interested in the chances of winning a lottery or inheriting a disease. What are the Odds? IV: Probability A probability is the number of expected successful events divided by the number of attempts. If you roll a single die the probability of rolling a five, P() = 6 since there is only chance of a being rolled out of 6 possible events. An event is an outcome such as being rolle The probability of an event happening such as a being rolled is written P(). ) If you roll a single die, what is the probability of rolling: An even number? An odd number? A number less than 3? 9) In a deck of cards, there are cards: 36 number cards, face cards, and four aces. Find P(face card). a. b. What are the Odds? V: Probability cont.... Information about probability: The probability of an event can be any number from 0 through You can represent probability as a: fraction, decimal, or percent If an event is impossible, it has a probability of 0 If an event is certain, it has a probability of A probability can t be greater than because an event can t be more likely than certain. Less likely than not Probability More likely than not % % 0% 7% 00% impossible unlikely as likely as unlikely likely certain 3 0) Represent each probability as a: Fraction Decimal Percent The probability of or Mendel (a 9 th century Central European monk now recognized as a famous genetics researcher) speaking at your school The probability of heads when flipping a coin P(Ace) when pulling a card from a deck Probability of year 00 following 00 Copyright 00. All rights reserve 00
5 Page of 7 Module #: Tutorial 0f: Probability Probing further into the subject of probability in genetics, the Probability Team ran across definitions that they had to understand first. What are the Odds? VI: Theoretical vs. Experimental Sample Space list of all possible outcomes of an activity. Spinner A has possible outcomes all equally likely:,, 3,. P() is and, 3, have that same probability.,, Spinner A 3, & represent the Spinner B sample space. Theoretical Probability: Predicted by a scientific hypothesis Without doing an experiment or analyzing data Ex: Predict how many times you would expect to land on heads if flipping a coin 0 times. Since P(heads) = (number of sides with heads) / (number of sides) =, then take P(heads) (number of flips). exactly half of the 0 flips. 0 = or Spinner B has members of its sample space:,, 3,,. These outcomes have different likelihoods. P() is, since there is only out Experimental Probability: Predicted by sampling Use of an experiment or collecting and analyzing data of choices that has the number. P() is or, since there are out of choices having the number. P(3) is also or. Ex: Suppose you flipped a coin 0 times and it landed on heads 6 times. 6 3 Then the experimental probability is 0 or. This is fairly close to the theoretical probability. The more you flip the coin in experimental probability, the closer it approaches the theoretical probability. ) What is the theoretical probability of spinning a Gray? a. Purple b. 6 ) Sam spins the spinner three times and all three times it lands on yellow. If Sam spins it one more time, what is the theoretical probability that it will land on yellow? a. 3 b. 3 Red Green Lt. Blue Gray Yellow Blue Pink 3) What is the theoretical probability that the spinner will not stop on green? a. b. 7 ) Which of the following is true? a. Experimental probability is predicted before experimenting b. Experimental probability is found by experimenting and collecting data Theoretical probability always has the same answer as experimental probability Theoretical probability is found by experimenting and collecting data 3 Copyright 00. All rights reserve 0
6 Page 6 of 7 Module #: Tutorial 0f: Probability The P Team felt they could look more closely at how this probability affects genetics. They finally understood how the two subjects intertwine after reading about or Mendel s ideas of the 60 s. What are the Odds? VII: More Genetics Up until Mendel s time, genetics was thought to follow a blending theory. Mendel observed that some traits do not show up in the next generation, and therefore the blending theory was not soun Mendel found: Inheritance of a trait is determined by genes or units passed down to a generation unchanged An individual inherits one such gene from each parent for each trait A trait may skip a generation and show up in the next generation DEFINITIONS: Alleles: variations of the same gene can be dominant(takes over), recessive(gives in) or codominant(works together) Genotype specific combination of alleles for a given gene Phenotype physical outcome of the combinations of alleles (or a result of the genotype) Another parent One parent Ex: The trait could be the color of a flower. The genotype could be the combinations: redred redwhite whitewhite The phenotype results in only two types since red is dominant and white is recessive: redred would result in red phenotype redwhite results in red phenotype whitewhite results in white phenotype Punnett Square: Invented by Reginald Punnett in early 0 th century Easiest way to calculate probability of inheriting a specific trait Graphical method to discover all possible G G Y YG YG Y YG YG combinations of genotypes Setting up and using Punnett Square: Draw a grid Place genotype of one parent across top and genotype of other parent down left side Copy row and column letters into empty square Calculate the probability or percentages There are four possibilities in a Punnett Square so the probabilities are out of : ) Given Y is dominant and G is recessive in the Punnett Square what is the probability of Phenotype Y for the next generation? a. b. Y Y YY G YG 6) Suppose two people were carriers for a disease such as SickleCell Anemia. Let s define A as the dominant normal allele and a as the recessive abnormal allele. The a is the one responsible for the sickness. The disease will only happen if there is an aa combination. Given the Punnett Square, fill in the empty grid and calculate the probability of the next generation having this disease. A a a. b. a 3 G YG GG Copyright 00. All rights reserve 0 3 a
7 Page 7 of 7 Module #: Tutorial 0f: Probability The Probability Team finished their research by studying probability of mutually exclusive events and the probability that two events will both occur. What are the Odds? VIII: Independent vs. Dependent Probability of one event with Mutually exclusive events: One or the other event can occur, but not both P (Event or Event ) = P (Event ) + P (Event ) Ex: If you have cards A, K, Q, J, 0 find P(A or Q) Mutually exclusive: if you pick one you can t pick the other P(A or Q) = P(A) + P(Q) = + = Probability of Two Events: Independent When the outcome of one event does not affect the outcome of another event P(Event and Event ) = P(Event ) P(Event ) Ex: If you have cards numbered A, K, Q, J, 0 and a coin find P(0 and heads) Two Events, Independent: one doesn t affect the other P(0 and heads) = P(0) P(heads) = = 0 Dependent When the outcome of one event does affect the outcome of another event P(Event and Event, given Event ) = P(Event ) P(Event, given Event ) Ex: If you have cards number A, K, Q, J, 0 find P(J and then 0) Two Events, Dependent: the second probability changes after the first occurs P(J and then 0, given J) = P(J) P(J, given 0) = (only possible choices left) = 0 7) Four scientists were chosen as equally important to the field of genetics by your friends, but only two will be given the highest honors in your group. Since all of you feel they gave equal contribution to the field, we must randomly pick two names. Find P(Mendel and Punnett, given Mendel) a. b. ) If you have cards A, K, Q, J, 0 and a coin, find P(not Q, and heads) a. b ) If you have 6 colored pens: Blue, Green, Yellow, Pink, Red, and Orange, find P(Blue or Pink) a. b. 6 b. 6 You can now join the Probability Team since you have become an expert in calculating many probabilities! The probability or odds of you having fun is very high! Copyright 00. All rights reserve 03
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