1 It’s Probably Probability By Virginia V. Lewis NSF Scholar

2 How do you play Pick 3? There are three containers. Each container has 10 balls in it labeled from 0 to 9. The player picks a three-digit number. If all three digits match in the correct order, you win $500

3 Let’s Play! Pick a three-digit number. Record your number on your class notes. Hope you’re feeling lucky

4 Important Terms →1 - implies the event is guaranteed to happen →0 – implies the event is impossible →.5- implies the event has a 50% percent chance of occurring Probability – the likelihood that an event will occur If the sample space is limited in size…. A Probability of :

5 Independent vs. Dependent Events Sometimes the outcome of an event will affect the next event and sometimes it will not. Independent events are two events where the outcome of the first event does not affect the outcome of the second event. Dependent events are two events where the outcome of the first event affects the outcome of the second event.

6 Chip Activity

7 Practice #1 Each letter of the word Lucky is written on a separate slip of paper. The five slips of paper are placed in a paper bag and two slips are drawn at random. The first letter is replaced before the second letter is drawn.  Is this problem an example of independent or dependent events?  Find the probability the first letter is L and the second letter is k.

8 Practice #2 Shae has four coins in her pocket. She has a dime, one penny, a nickel, and a quarter. Shae needs twenty-six cents to pay the cashier. She reaches in her pocket and takes out two coins.  Is this situation an example of independent or dependent events?  What is the probability that she pulled out twenty-six cents?

9 Card Activity

10 Practice #3  Calculate the theoretical probability of drawing out a spade and then another spade with and without replacing the first card.  How is this theoretical probability related to the experimental probability the class determined?

11 Probably Hereditary Probability plays an important role in the area of genetics. What do you think the probability is that a child about to be born will be male? What if the mother already has two boys? Three boys? Let’s take a look at the genetics and see……

12 Basic Genetics When we are born we have one chromosome from our mother and another chromosome from our father. If you are a girl than your chromosomes are XX and if you are a boy then your chromosomes are XY.

13 Punnett Squares Punnett squares are used to determine the possibilities of inheritance when a male and female reproduce. On the left side we put the traits of the mother and on the top we put the traits of the father. XY X X Father’s Chromosomes Mother’s Chromosomes

14 How many of these possibilities will be male children? Female? What is the probability of a child being male? Female? XY X1X1 X1XX1XX1YX1Y X2X2 X2XX2XX2YX2Y Now we take one chromosome from each parent and fill in the chart. There are four possibilities of chromosome patterns for a new baby.

15 Tree Diagram Tree diagrams are often used in math class to find the probability of an event. The tree diagram for the gender problem is shown on the next slide.

16 X1X2X1X2 X Y X Y X1XX1X X1YX1Y X2YX2Y X2XX2X Sample Space Mother’s Chromosomes Father’s Chromosomes

17 Probability of Gender  Is the birth of each child a dependent or independent event?  Does the gender of the first child affect the gender of the second child?  Determine the probability that if a family has two children they both will be male.

18 More about Heredity Each of us inherits an allele for a certain trait from each parent. These alleles combine to form our genotype. Our genotype determines our phenotype.

19 My Mother My mother has brown eyes. Her alleles are B and b. Her genotype is Bb. Her B allele is the dominant brown eye trait. Her b allele is the recessive blue eye trait. Since she has one dominant and one recessive allele she has the dominant brown eye trait.

20 My Father My father has blue eyes. His alleles are b and b. His genotype is bb. His b alleles are both recessive. Since he has two recessive alleles he has the recessive blue eye trait.

21 So What About Me? I have one allele from my mother and one allele from my father. My genotype is determined by these alleles. I will have brown eyes if my genotype is BB or Bb. I will have blue eyes if my genotype is bb.

22 Let’s Use the Punnett Square bb BBbBbBbBb bbb There are four possible outcomes for my genotype. The possibilities are Bb, Bb, bb, and bb. Two of these four genotypes mean that I will have brown eyes. I have a 50% chance of having brown eyes. Mother’s Genotype Father’s Genotype

23 BbBb b b b b BbBb BbBb bb Sample Space Mother’s Genotype Father’s Genotype Now Using a Tree Diagram! Wow! The results are the same. I still have a 50% chance of having Brown eyes.

24 Practice #4a Use a Punnett square to determine the probability I will have freckles.  F is the dominant allele for freckles and f is the recessive allele. If I have a genotype of FF or Ff I will have freckles. If my genotype is ff I will not have freckles.  My mother’s genotype is FF and my father’s genotype is ff.

25 The Solution ff FFfFfFfFf FFfFfFfFf I have four possible genotypes. But they are all the same, Ff. I have a 100% chance that I will have freckles. Guess I better stay out of the sun Mother’s Genotype Father’s Genotype

26 Practice #4b Now try making a Tree Diagram for this situation. What is the probability I will have freckles?

27 FFFF f f f f FfFf FfFf FfFf FfFf Sample Space Mother’s Genotype Father’s Genotype The Solution The same results!!! I will definitely have freckles

28 Back to Pick 3 Is the Pick 3 Game an example of independent or dependent events? Use your new knowledge of probability to determine the likelihood of winning $500 in Pick 3. Now let’s play!!!!!!!!!!!!!!!!!!!!!!