Probability
Lesson 44
In the warm-up for lesson 44 your students will find the products of fractions. Its important that they know how to multiply fractions because they will need to be able to do this to find compound probabilities. Penny has a bag of marbles. There are 4 green, 7 blue, 5 white, and 4 black marbles. Penny is going to reach into the bag and choose a marble without looking. She is then going to put the marble back in the bag and choose another marble without looking. What is the probability that she will choose a blue and then a black marble?
We are going to study this problem, our first step in "S" is to underline the question. What is the probability that she will choose a blue and then black marble? This problem is asking me to find the probability that the marble she chooses will be blue, and then black. When we are working with probabilities most of the time we begin by representing our probability as a fraction, such as 1 over 3, or one-third. We can convert fractions and probabilities into decimals and percents. If we want to convert the fraction one-third into a decimal we will first begin by dividing 1 by 3. 3 does not go into 1 at all. Not evenly so we use a zero and we have to add a decimal. We add another zero, does 3 go into ten? Yes it does. It goes 3 times. 3 Times 3 is 9. And 10 minus 9 is 1. From here we ask ourselves does 3 go into 1? It doesn't so we add another zero and bring it down. Does 3 go into 10? Yes... 3 times. 3 times 3 is 9 and we subtract and get 1. We will continue to do this over and over. Add another 0, bring it down. Does 3 go into 10? Yes. 3 times. With 9 leaves us a remainder of 1. Because one-third is equal to point 3 repeating. We can represent the decimal for one third as 0.3 with the bar, and the bar tells us that the 3 continues to repeat. If your students do not have the bar it is wrong. We can also convert our decimals to a percent.
We know that 0.3 repeating means 0.3333 continues to move on.
If we represent our decimal as a percent we have to move the decimal over 2 places to the right. 1, 2, we know that we continue to have 3's here and so our decimal becomes 33.3 repeating, percent. The fraction one third is equal to point 3 repeating and 33.3 repeating percent.
If we want to convert the fraction one-eighth we once again have 1 divided by the denominator of 8. 8 cannot go into 1 so we must add a decimal does not go in there. But 8 can go into 10, 1 time. 1 times 8 is 8 then we subtract and we're left with the remainder of 2. If I add another 0 eight does go into 20 twice. 2 times 8 is 16. When I subtract I'm left with the remainder 4. 8 does not go into 4 so I have to add another 0 but 8 will go into 40, 5 times. And 5 time 8 is 40. Our decimal one-eighth is equal zero point 125. and if we want to convert the decimal 0.125 to a percent then we move our decimal to the right 1, 2 places to create 12.5 percent.
If we discuss theoretical probability, your students have to understand that we are guessing what will happen. We know what the possibilities are and we know what we want to happen, but we have not actually done the experiment. If we are trying to find the theoretical probability it is the probability of an event which is the number of favorable outcomes over the total number of possible outcomes. If we said what is the probability of spinning 1 on this spinner, we only have 1 one. So the number of favorable outcomes is 1 over the total number of possible outcomes we have 1,2,3,4 possible outcomes. So the probability of spinning will be 1 out of 4. We haven't spun the spinner and it landed on 1. But we're saying it could happen and there is a 1 in 4 chance that it will land on 1.
If we said what is the probability of the spinner landing on an even number 2 and 4 are both even numbers. That would mean that we would have 2 favorable outcomes divided by the possibility of 4. So two-fourths or one-half. We look at experimental property using the table. Jordan uses the table below to record his last 20 spins. We have sections 1,2,3 and 4, these are our possibilities. If he spins the spinner 4 separate times it landed on 1, 3 times it landed on 2, 5 times it landed on 3, and 8 times it landed on 4.
Experimental probability deals with what actually happens. The trial took place and we recorded what happened during the trial its not a what could happen its a what did happen. The probability of an event is equal to the number of times an event occurs over the total number of experiments. In other words how many times did it land on 1 compared to the number of times we actually spin the spinner. If we wanted to know the experimental probability of the spinner landing on 3 it actually landed on 3, 5 times out of the total of 20 spins. So our experimental probability would be 5 out of 20. If we wanted to know the experimental probability of the spinner landing on an even number 2 and 4 are both even so 3 plus 8 is eleven. So the probability of it landing on an even number is 11 out of 20.
When only one event is occurring it is considered simple probability. Such as spinning a spinner 1 time or rolling a number cube one time. Sometimes we want to know what the probability of 2 events happening at the same time is. This is considered compound probability. There are two different types of compound probability. 1 involves independent events. When we are working with independent events the probability of independent events such as A and B is equal to the probability of A times the probability of B. Your students are going to do a trial where they flip a red and yellow chip and they roll a number cube. They are going to record what happens when they flip the chip and roll the number cube. If I flip the number chip and it rolls on red and then I roll the number cube and it lands on 2. Red 2 is one of my possibilities and I will put a tally mark here.
We have several possibilities when we split the chip and roll a number cube. The chip could either land on red or yellow. This gives us 2 possibilities. Once the chip lands on red the number cube could land on 1,2,3,4,5 or 6 so here are 6 possibilities and if the chip lands on yellow we have 1,2,3,4,5, 6 more possibilities so there are a total of 12 possibilities that your students have that could roll the number cube could land on 6 and the cube would be yellow. If we wanted to find the possibility of the chip landing on red or the probability of the chip landing on red and rolling a 3, we could also use our formula's. The probability of the chip landing on red it could be either red or yellow so there are 2 possibilities and only 1 of them is red the probability of a 3 there are 6 numbers and only 1 of them are 3 and when we multiply 1 times 1 we get 1 and 2 times 6 we get 12. This makes sense because out of our 12 possibilities only one of them is red and 3.
Another type of compound probability deals with dependent events. When we are working with dependent events it is very important to know what is taking place. The probability of "A" and then "B" is equal to the probability of "A" Times the probability of "B" after "A." Debra has a bag of marbles the bag has 5 blue marbles, 4 red marbles, 7 green marbles, and 4 yellow marbles. Dependent events deal with situations like this where Debra can either pull out a marble and then put it back in the bag, or pull out a marble, leave it out and choose a second marble out of the bag. If we wanted to define the probability of her picking out a yellow marble not replacing it and then picking out a green. Then we find the probability of "A" which is picking out a yellow marble.
We have a total of 20 marbles in the bag and 4 of them are yellow. So the probability of yellow leaving it out and then green. The probability of yellow are 4 out of 20, time the probability of green after we took out the yellow. If we take one yellow marble out we only have 19 marbles left in the bag. Out of those 19, 7 are green. We will multiply these two fractions to get 28 over 380 which reduces to 7 over 95. Dependent events can also work with replacement. If we wanted to find the probability of her picking a blue marble replacing it and then picking another blue marble. The probability of a blue marble is 5 out of the 20, If she replaces that marble back into the bag we still have 20 marbles and we still have a chance to get 5 blues. This leaves us with the probability of 25 over 400. Which simplifies to 1 over 16. You have to make sure when you're working with dependent events if what happens deals with replacement if something is replaced or whether it is not replaced it is left out and then you pick again. To make our foldable we will begin by folding our paper vertically. I call this a hot dog fold, we will leave it folded and then fold it horizontally or a hamburger fold and then another hamburger fold. When we open this up we have created 4 rectangles. We will pick up the top half and cut down to the center fold from each crease.
We are creating our 4 flap. And we will use these 4 flaps to represent experimental probability, theoretical, sample space, and compound. Your students will then complete the inside where they will have an example and any other information. For example sample space and independent and dependent.
We have already asked the problem so we know the problem is asking you to find the probability that the marble she chooses will be blue and then black. To organize the facts we will strike our facts and decide if they are necessary or unnecessary. Penny has a bag of marbles, some students will say this is a necessary fact and they won't want to cross it out, Some students will say it is unnecessary and they will cross it out. Some of them will not want to write it down. I think it is necessary but I'm not going to write it down because, we are going to, we know she has a bag of marbles. There are 4 green, necessary, 7 blue necessary, 5 white necessary, and 4 black marbles, necessary. Penny is going to reach into the bag and choose a marble without looking, this is necessary because she is going to pick 1 marble. She is then going to put the marble back in the bag, this is also necessary because she replaces the marble and she is going to choose another marble without looking because she picks the second marble. In L line up a plan we must choose an operation or operations and write in words what our plan of action will be.
Because we are going to find the probability we have to know the total number of marbles in the bag and then we are going to use multiplication because this is a dependent compound probability question. Our operations will be addition, and multiplication. To write in words what our plan of action will be, we will add the total number of marbles for the denominator we will find the probability of getting blue, we will find the probability of getting black, and multiply the 2 probabilities. In V, Verify our plan with action, we will first estimate our answer we know its a probability so it has to be somewhere between 0 and 1. Because there are not a significant amount of blue they're not larger than all of the other marbles put together or there aren't a significant amount of black meaning all of the marbles are black, then all of our guesses of probabilities will probably be less than one-half. We will first add our total number of marbles 4 plus 7 is 11, plus 5 is 16 plus 4, 20. so our denominator will be 20. the probability of blue, there are 7 out of the 20 blue the probability of black, there are 4 out of the 20 which are black which simplifies to one fifth and when we multiply seven-twentieths times one fifth we end up with the total probability of 7 over 100. In E examine your results does your answer make sense we go back to our question which asks us to find the probability. Probabilities are normally given as fractions so our answer makes sense. Is your answer reasonable? We said it would be somewhere between 0 and 1 and probably less that one half so yes our answer is reasonable. And is your answer accurate have your students check their work on another sheet of paper or use a calculator. We will now write our answer as a complete sentence. The probability of Penny getting a blue and then a black is 7 out of 100 or 7 percent. To close the lesson we will review the essential question.
Number 1: How do you find the experimental probability of an event. To find the experimental probability you will find the number times the event happened and this will be your numerator, and then you will find the number of trials conducted and this will be your denominator.
Question 2: How do you find the theoretical probability of an event. We will first find the number of favorable outcomes and that will be the numerator and we will divide by the total number of outcomes and that will be your denominator. In question 3 what is the difference between dependent and independent probability: Dependent probability means that the chance of the second event happening are dependent on the first event that happened. Independent means that the first event does not effect what happens in the second event.