Once again, The Law ofOR says, ifyou need to know the probability that one thing 0R another will take place, just add their separate probabilitiesof course its not (quite) that simple. Here's the fine print: the events in question have to be mutually exclusive In other words, you will win one prize, but not two, or three, or four. Here are some mutually exclusive events: - Your friend is pregnant OR not pregnant; as we all know, there's no such thing as a little pregnant. - Your mother offers you cake OR a brownie for dessert. You can't get both! - You have one dollar. You can buy fries OR a donut OR a small milkshake, but each one costs $1.00, so after you buy one thing, your money's gone, you're out of luck. Here are some nonmutually exclusive events: 0 Your grandmother offers you cake OR a brownie 0R both. Take both dear! - Your friend has a cold OR a flu OR maybe both, she looks so miserable. - You just got paid, to celebrate you could get fries OR a donut OR a small milkshake, 0R any combination of the above. Which of the following sets of choices are non-mutually exclusive? C] The chance of rain or snow in the weather forecast. C] the likelihood you left your keys in your back pocket, or your backpack, or on the counter. [:1 the chance that your father or mother has curly hair. [1 the light is either on or off. The Law of And So. how do we determine probability when we have two non-mutually exclusive events? We can use The Law of And, also know as the Multiplication Flulell Let's say our game show has reached rock bottom financially and now has to reduce the probability of contestants winning any prizes. To do this, our top three prizes (washer-dryer, toaster, and vacation) will have a certain probability of being selected as the big prize. The big prize will be placed behind one of three doors. and the other two doors will have pet rocks as a prize. Door!\" Doorvlz Doom: What is the chance of winning the vacation given a contestant has 1 in 3 chance of selecting the doorwith the big prize, and the big prize has am) 8% chance ofbeing the toaster. a(n) 14% chance of the washer-dryer, and am) 6% chance of the vacation? (ignore that the percentages don't add up exactly to 100% - not important for this question) . Report your answer as a percent to two decimal places, eg, 1.02%, not 0.00102. - When convening 1/3 to a percent. round to two decimal places. - Do not include the percent sign when reporting your answer. We use The Law of And, also known as The Multiplication Rule, when two things occur simultaneously. When you want two things to happen atthe same time, you multiple the probabilities of the individual events to nd out what the probabiilty of both events happening simultaneously is. Written symbolically, The Law ofAnd looks like this: P(E1 and 52) = P(E1) * P(E2) Let's say that the game show has adopted a more complicated probability structure. There is am) 79% chance the big prize will be a toaster, am) 45% chance the big prize will be the washer dryer, and am) 87% chance the big prize will be the vacation. If the game show puts the big prize behind Door #1 25% of the time, behind Dannie 25% of the time, and behind Door #3 50% ofthe time, then what are the chances of winning the toaster oven if a contestant picks Door #1? Given: - P(E1)= 79% - P(E2)= 25% - P(E1 and E2) = P(E1) * P(E2) Find: P(E1 and E2) Report P(E1 and E2) as a percent rounded to two decimal places. In order for the Law ofAND to work. the two events have to be independent. What does that mean? It means that one event does not influence the other. The fact that one event has occurred does not make the other more likely. For example, getting hurt and going to the hospital are NOT independent events. My chances of getting hit on the head by a falling tree this year are maybe 1 in a hundred, or 1%. My chances of going to the emergency room this year are maybe also 1%. So using the Law of AND, my chances of getting hit in the head would be1% * 1% = 0.01 * 0.01 = 00001 = 0.01%, or one-hundreth ofa percent. But in fact, IF I get hit in the head, I'm likely to want to go to the hospital for that very reason. So, the two events are NOT independentll And my chances of getting hit in the head and going to the hospital are much bigger than 0.01%. Maybe more like 0.5% (about half the time that I get hit in the head I go to the hospital). Which of the following events are independent? C] Flipping a coin once and getting tails, and ipping a coin a second time and getting heads. C] Pulling one card from a deck of cards and getting a red suit (hearts or diamonds) and pulling a second card from the same deck of cards and getting a second red suit. C] The probability of it rainingon the same day there is a pop quiz in one ofyour classes. C] The probability of being hungry and the probability of getting a snack. Let's say the weatherwoman says the chance of rain is 60% and the chance of snow is 50%. What's the chance of rain OR snow? If you just add, you get: P(rain or snow) = P(rainl + P(snow) = 110% Probabilities over 100% are meaningless. So right off the bat you know something is wrong. How about the chance at rain AND snow? The Law of AND would say P(rain and snow) : P(rain) * P(snow) = 30% But we all know that rain often turns into snow and snow turns into rain, so these two events are not independent either and we can't use the multiplication rule. In fact, we really can't say ANYTHING about rain combined or not combined with show. We can't add rain OR snow, but we also can't multiply to get rain AND snow. The two events aren't mutually exclusive. and they aren't independent. Neither law does us any good. In this case what you would need to know is the conditional probability in order to determine the probability ct rain. it it is already snowing. or snowing. if it is already raining. Conditional Probabilities are usually written as P(A| B) and read aloud as "The Probability ofA, given B." If there is a 60% chance ofsnow, and a 14% chance of rain, if it is already snowing, then we can let snow be Event A, and rain be event B. Therefore, P(A) = 60, and P(B|A) = 14. To find the chance of it raining, after it has started snowing, we can multiply P(A)*P(B|A) to find the answer. What is the probability of it raining, P(B), if it has started snowing? Report P(B), given that it has already started snowing as a percent. Round to the nearest whole percent (for example, 30 rather than 30.4). Question 6 1.1 pts Given a bag with 40 red marbles, and 53 blue marbles, what is the probability of pulling two consecutive blue marbles? Report your answer as a percent rounded to two decimal places. Question 7 0.8 pts In order to calculate probabilities you must know how the different pieces (terms) are used. Match the term with the appropriate definition. denominator in a ratio V [ Choose ] the number of occurrences of an event the occurence of the thing in which you are interested event the set of all possibilities counting of all the possible outcomes numerator in a ratio the extent to which an event is likely to occur sample space [ Choose ]Question 8 0.8 pts In the Frequentist Interpretation of probability, the frequency of an event is equal to the total size of the sample space. O True O False Question 9 0.8 pts Which of the following calculations require that you utilize the addition rule? O Calculate the probability of children with both cystic fibrosis and polydactyly when parents are each heterozygous for both genes O Calculate the probability of black offspring from the cross AaBb x AaBb, when B is the symbol for black O Calculate the probability of each of four children having cystic fibrosis if the parents are both heterozygous. O Calculate the probability of purple flower color in a plot of 50 plants seeded from a self-fertilizing heterozygous parent plant O Calculate the probability of a child having either sickle-cell anemia or cystic fibrosis if parents are each heterozygous for both Question 10 0.8 pts Which of the following occurrences are you looking at in a union of two events? O the occurrences of event 1 the occurrences of event 2 the occurrences of both events 1 and 2 together (All of the occurrences of event 1 plus all of the occurrences of event 2) O occurrences of event 1 in concert with occurrences of event 2 (when they happen together)Question 11 0.8 pts If the probability of one event has no influence on the probability of the other then the events are said to be: O conditionally dependent O independent O complementary O mutually exclusive Question 12 0.8 pts What is the sample space when picking just one marble out of the bag? O S = {Blue; Red} O E = {Blue; Red} O S = {Blue; Blue; Red; Red; Red} O E = {Blue; Blue; Red; Red; Red}Question 13 0.8 pts Now, lets calculate the probability of pulling out a blue marble, followed by a red marble, using the multiplication rule, and conditional probabilities! Based on the image below, we know that P(A) = 215 and P(B|A) = 3/4. Our formula will be P(A and B) = P(B|A) * P(A) Report P(A and B) as a decimal rounded to two places Question 14 0.8 pts . / O 0.. \\ Q . .0. / 5 :.. k~.'2'~>~' 4 C u 33' I"I *I .5 Using the same conditional probability tree as before, what are the chances of pulling two red marbles consecutively? Report your answer as a decimal. We are going to start with the two most important laws of probability When two events can't happen at the same time, we use "The Or Law" and we add the probabilities together. That's why there is a summation symbol (X ) in the photo for "The Or Law'h When two or more things happen simultaneously, we use "The And Law" and we multiply their probabilities together. That's why there is a product symbol ( |'| ) in the photo for "The And Lawi" Let's say you are on a game show, with several fabulous prizes: 3 5speed automatic washer-dryer, a deluxe selfcleaning toaster oven, a oneweek allexpenses-paid tropical vacataion, or the consolation prize of 2dozen pet rocks Your chances of winning are as follows: E Chances Toaster 14% Wash er- dryer 6% Vacation 1 % Pet Rocks 90% What are the chances of winning an appliance (ie. the toaster or the washer-dryer)? Report answer as a percent, not as a decimal; In other words, if it's 55%, report it as 55, not 055. The Law of Dr The chances of one event OR another event occurring = the sum of the chances of each event separately. Let's call the rst event E1, and the second event E2. In this case, let's make E1 the probabilty of winning the toaster, and E2 the probability of winning the washer-dryer. We can now abbreviate these P(E'l) and P(E2) for the probability of event 1, and the probability of event2. lfthe game show runs another contest where you can win the prize by picking a ticket out ofa hat and your chances of winning are listed in the table below. The probabilities in this problem are given as percentages. That means if the hat contains one hundred pieces of paper in it, then it will have 19 pieces of paper labeled "toaster", 3 pieces of paper labeled "washer-dryer", .7 pieces of paper labeled "vacation" and 71 pieces of paper labeled "pet rock". If the game show ran the contest again using the following probabilities, what would be the chances of winning the tropical vacation? Prize Probability Toaster 19% Washer-d ryer 3% Vacation ? % Pet rock 71% Report answer as a percent, not as a decimal. In other words, it it's 55%, report it as 55. not 0.55 Question 17 0.8 pts The Rule of Or continued. If the game show ran the contest a third time and advertised an 1% chance of winning the vacation by pulling a piece of paper out of the hat, and there were 200 pieces of paper in the hat, how many pieces of paper would have "vacation" written on them