Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Level F Lesson 30 Simple Probability The objective for lesson 30 is the student will investigate simple probability. We have three essential questions that will be guiding our lesson. Number 1: What is simple probability? Number 2: How do I use probability in everyday life? And Number 3: What is the difference between theoretical and experimental probability? Our solve problem for this lesson is: Davida has the following spinner. Her friend, Eva, has a spinner with 6 numbers on it. If Davida spins the spinner, what is the probability she will spin a 2? We’re going to start by completing the S step for Solve. We’re going to underline the question. What is the probability she will spin a 2? Then we’re going to complete the statement: this problem is asking me to find the probability that Davida will spin a 2. We’re going to be creating a foldable today that we can use to organize our information about sample space and simple probability. We’re going to start with our paper laid horizontally on our desk and we’re going to fold down from the top leaving just a small space at the bottom. We’re going to fold and crease. We want to fold the paper in thirds so we’re going to fold vertically and then bring our top down. We’re going to open that back up. On the section that has the shorter edge, I’m going to take some scissors and I’m going to cut to the fold. The last step I’m going to take the flap at the bottom and I’m going to fold that up. On one of the sections I’m going to write sample space. When you open your foldable on this part you’re going to include information about sample space. For sample space you can: make a list, make a tree diagram, make a chart, draw a picture, or use the Fundamental Counting Principle. The question for sample space is: How many outcomes are there? On the remaining space you’re going to include an explanation of sample space. There’s a list and what we’ve done is create this list and the tree diagram using the possible outcomes from flipping a coin. If we have two coins to flip we could have a: head and a head, or a head and a tail or we could have a tail and a head, and a tail and a tail. So if our first outcome was either heads or tails each of those would have 2 outcomes. Here’s a chart we’ve created showing those outcomes and we’ve drawn a picture. The last thing we’ve done is explain it using the Fundamental Counting Principle. There are 2 possible outcomes for our first event, 2 for the second and when we multiply those, we have a sample space of 4. We’re going to begin by discussing some of the vocabulary that we use with probability. I have four words on my chart here that are some of the things we may say when we’re talking about probability. We have likely events, unlikely events, certain and impossible. When we think about situations, we can categorize them in one of those four ways. Let’s take a look at likely. If you are school, what is the probability that you will eat at lunch at school on a school day? That’s a likely event because it will probably happen. Most of the students who are at school will eat their lunch. Is it likely or unlikely that you will stay overnight at school? It’s probably an unlikely event because only in a very rare case, such as a special event for a club, 1 would you ever be staying overnight at school. So you would probably not stay overnight. That’s an unlikely event. Is it certain that the sun will rise tomorrow? This is probably a certain statement as the sun rises each day even when we don’t see it. How about an impossible event? The temperature outside today will be 150 degrees. It’s probably impossible because that temperature does not occur here on earth. You may think of some other events and discuss those with your teacher about things that would be likely, unlikely, impossible and certain. We’re going to be talking about simple probability with a fair number cube. On our page we have an example of a fair number cube that would have the number 1 through 6 on the sides. There are six possible outcomes if we were to toss that number cube: 1,2,3,4,5, and 6. Each of those outcomes is equally likely or has a fair chance of being tossed. If the number cube is tossed it has an equal probability of landing on each of the sides. In example 1, we’ve drawn a picture of each of the possible outcomes for the fair number cube toss. What is the probability of tossing a 4 on the fair number cube? If we were to toss the fair number cube, this is the 6 possible rolls we could have, it could be a 1, a 2, a 3, a 4, a 5, or a 6. However, only one of those rolls is a 4. So, there is one 4 in the possible rolls. The probability of rolling a 4 is 1 out of 6. The probability can be written as 1 to 6, 1 colon 6 or one-sixth with a fraction bar. In example 2, we’re using the same sample space and our question is: what is the probability of tossing and even number on the number cube? Out of our choices, 3 of those numbers are even. 2, 4 and 6 are even numbers. The probability of rolling an even number is 3 out of the 6. This probability can be written as 3 to 6, 3 colon 6, or the fraction 3 over 6. This ratio of three-sixths can be simplified to one-half. In example 3, we still have our same number of possible outcomes and our question is: What is the probability of tossing a number greater than 2. There are 4 numbers in our sample space, 3, 4, 5, and 6 that are possible rolls that are greater than 2. The probability of rolling a number greater than 2 then is 4 out of the 6. The probability can be written as 4 to 6, 4 colon 6, or the ratio four-sixths. The ratio of 4 over 6 can be simplified to 2 over 3, or two-thirds. We’re going to take a look at simple probability theoretical. We’re going to start. Our problem is we’ve been give two cubes of different colors. We have a green and a red. So the first thing I’m going to do is I’m going to take my green marker and shade the representation of that and I also have a red cube. When we have a theoretical probability it says: What is the probability of drawing one color. All the possible colors are called the possible outcomes. In this case there are 2 possible outcomes: green and red. If I were to pick one of these cubes without looking I would choose either a green or a red. If I were to predict green then the event would be: choosing green. If you predict red, then your event would be to choose red. There are two possible outcomes: green and red. This theoretical probability of an event is a ratio written as the number of favorable outcomes over the number of probable outcomes. In the case of our example, the probability of choosing green, which is the event, is written 2 as a ratio as the number of green cubes we have in the envelop, which was 1, over the total number of cubes, or one-half: 1 green and 2 green and red. What does this mean? It means the probability of choosing a green cube from the envelope is 1 out of 2. In our second example, we’re going to have 3 cubes: 2 of one color and 1 of the second color. So, we have 2 reds and 1 is green and we’re trying to find the probability of drawing the first color. First thing I’m going to do is shade in to represent a pictorial example of my three number cubes. If I put my 3 cubes in an envelope, 2 reds and a green, what color will I pull out if I don’t look? I would pull our either a red or a green. I would be more likely to pull out a red because there are 2 reds in the envelope and only 1 green. If you predict I will pull a red out of the envelope then the event would be choosing red. If you predicted green then your event would be choosing green. There are still only 3 possible outcomes: red, red, or green. So, we’re going to use our formula for theoretical probability to determine the probability of choosing a red cube. The probability of red is the number of red cubes over the total number of cubes. We have 2 red in the envelope and a total of 3 cubes. What does this mean? The probability of choosing a red cube from the envelope is 2 out of 3. We’re going to move on and talk about experimental probability. I’ve left my information from theoretical probability but we’re going to talk about experimental probability. Experimental probability is where we’re going to actually have to pull the cubes from the envelope and base our probability on the results of our experiment. We still have a green and a red cube. We’re going to shade those in and if we place those cubes in an envelope we’re going to do an experiment. What you’re going to do is, with your partner, you’re going to put the cubes in and without looking draw a cube. You’re going to create a tally chart here in this box that indicates how many times you drew each color. You’re going to do 20 events or 20 tries of pulling a cube with your partner. Ok, I have completed the experiment by pulling a cube without looking 20 times from my envelope. Remember, your results will differ from mine because each experiment will be different. I had 8 times that I pulled a red and I had 12 times that I pulled a green. What’s the difference between the experimental and the theoretical probability? In the theoretical probability, we base our probability on the favorable outcomes over the possible outcomes. In the experimental probability we base it on the favorable outcomes over the number of trials in our experiment. In the case of my experiment, we said green; I had 12 times that I drew a green out of 20. That can be simplified to 3 over 5. The question is: why is my experimental probability not the same as my theoretical? What happens is if the number of experiments is increased; the experimental probability will get closer to the theoretical probability. The simple probability example problem that we’re going to use to talk about theoretical and experimental probability is this: Given the numbers 2, 3 and 4, what is the probability of getting a 3? I want you to take a simple index card and we’re going to cut that into 3 sections. After we do that, we’re going to number the sections: 2, 3 and 4. Those numbers then represent our sample space. To determine 3 the theoretical probability of drawing a 3 remember we have our 3 number cards in our envelope: a 2, 3 and 4. There is only one 3 in that problem. So, the number of 3’s is 1 and the total number of numbers that are possible to draw is 3. What does this mean? It means that the probability of choosing a 3 from the envelope is 1 out of 3. For our experimental probability we have the number of times that a favorable outcome occurs over the number of trials in the experiment. We’re going to complete 20 trials again and we’re going to draw the number cards without looking 20 times. I’ve just finished drawing my number cards, 20 separate trials or experiments, and I’ve drawn a 3 seven times. That means my experimental probability of drawing a 3 is 7 out of 20. We’re going to be completing the foldable for probability. We’ve already done the sample space section. Now we need to label the middle section theoretical probability and the third section experimental probability. When I open my theoretical probability flap, on this section underneath the title, we’re going to include this information: What do you think would happen when…? And this is our formula for the probability. The event is equal to the number of favorable outcomes over the total number of possible outcomes. On the inside of the theoretical probability folder you’re going to include an example of tossing a coin. There are 2 sides which are possible outcomes. So, the denominator is 2. One side is heads, so the numerator is 1. The probability of heads is one-half. In the last section of the probability foldable, underneath the flap where you have written experimental probability you’re going to write this: See what happens when you do it! To remind to that in experimental probability you need to perform an experiment. The probability of an event is the number of times the event occurs over the total number of experiments. In the last part of your experimental probability foldable you’re going to include this information. The example is tossing a coin. If you tossed a coin 20 times, your denominator is 20. If you landed on heads 7 times, your numerator is 7. Your probability of heads in this experimental probability was 7 over 20. We’re now going back to the solve problem from the beginning of the lesson. Davida has the following spinner. Her friend, Eva, has a spinner with 6 numbers on it. If Davida spins the spinner, what is the probability she will spin a 2? We’ve already completed the S step by underlining the question and completing the statement: the problem is asking me to find the probability that Davida will spin a 2. We’re going to move to the O step and we’re going to identify the facts. Davida has the following spinner. Her friend, Eva, has a spinner with 6 numbers on it. If Davida spins the spinner, what is the probability she will spin a 2? Then we’re going to eliminate the unnecessary facts. We have a picture of the spinner and we know she has that. Her friend, Eva, has a spinner with 6 numbers on it. We don’t need to know that either. Now we’re going to list the necessary facts. The spinner that she has has 4 sections labeled 1, 2, 3, and 2. Davida will spin one time and her event is a 2. Now we’re going to line up our plan. We’re going to choose an operation, which is N/A 4 meaning not applicable for this problem. Write in words what your plan of action will be. We’re going to determine the favorable outcomes and the possible outcomes and we’re using the formula for theoretical probability. The probability of the event is the number of favorable outcomes over the number of possible outcomes. Now we’re going to make an estimate. Our estimate is about 2 out of 4. Then for V, we’re going to carry out our plan. Our favorable outcomes are 2. Our possible outcomes are 4. The probability of spinning a 2 is 2 out of 4 or 2 out of 4 or one-half. We’re going to move to the E step now. Does your answer make sense? Yes, because we found the probability of spinning a 2. Is your answer reasonable? Yes, because 1 out of 2 or 1 over 2, one-half matches my estimate of 2 out of 4 or two-fourths. Is your answer accurate? Yes. Write your answer in a complete sentence. The probability that Davida will spin a 2 is 2 out of 4, or two-fourths, which can be simplified to one-half. Now we’re going to go back and answer the essential question from the beginning of our lesson. What is simple probability? How likely something is to occur. How do I use probability in everyday life? Prediction of the weather may be one possibility and the other examples that were discussed at the beginning of your lesson. What is the difference between theoretical and experimental probability? Theoretical probability determines the probability of an event using a formula. Experimental probability determines the probability of an event by doing an experiment. 5