O O We use The Law of And, also known as The Multiplication Rule, when two things occur simultaneously. When you want two things to happen at the same time, you multiple the probabilities of the individual events to find out what the probabilty of both events happening simultaneously is. Written symbolically, The Law of And looks like this: P(E1 and E2) = P(E1) * P(E2) Let's say that the game show has adopted a more complicated probability structure. There is a(n) 79% chance the big prize will be a toaster, a(n) 45% chance the big prize will be the washer dryer, and a(n) 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 Door #2 25% of the time, and behind Door #3 50% of the 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 of AND 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 be 1% * 1% = 0.01 * 0.01 = 0.0001 = 0.01%, or on e-hu nd reth of a 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 independentil And my chances of getting hit in the head and going to the hospital are much bigger than 0.01%. Maybe more like 05% (about half the time that I get hit in the head I go to the hospital). Which of the following events are independent? :I Flipping a coin once and getting tails, and ipping a min a second time and getting heads. D 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. D The probability of it raining on the same day there is a pop quiz in one of your 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 offthe hat you know something is wrong. How about the chance ot rain AND snow? The Law of AND would say P(rain and snow) = P(rain) ' P(snow) = 30% But we all know that rain otten 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 snow. 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 0' rain, if 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 of snow, 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). :i In the Frequentist Interpretation of probability, the frequency of an event is equal to the total size of the sample space. O True O FalseWhich 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 bothWhat 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) = 2/5 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. 0.3w/ 2 Using the same conditional probability tree as before, what are the chances of pulling two red marbles consecutively? Report your answer as a decimal. 0.3