E36: FTCE | General Knowledge | Mathematics | Calculating Probabilities of Coins and Marbles

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Welcome to episode 36 of FTCE seminar, a teacher certification podcast. I'm your host Mercedes Musto. Today we will be having a math conversation about probability. This is going to help you prepare for the FTCE general knowledge mathematics subtest. Hey, check out that last episode to review statistics. But for today, we're going to be diving into the concept of probability. What's probability and will understanding a little bit about probability help me solve real world problems? Probability? Probably! That's just fun to say like five times real fast. All right, let's get started. Probability is all around us.

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You may be familiar with probability in real life when you know you're trying to predict the weather or you may even hear yourself say something like, well, there's a 76% chance of rain through Thursday. Oh, that's great, yeah, I love the rain. The FTCE testing blueprint says, we need to understand counting procedures, tables and tree diagrams to solve real world probability problems. So let's define these terms.

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Number one, counting procedures involve systematically counting the number of possible outcomes in a given scenario. This method is useful when each outcome is equally likely. See, you'll apply this method in those questions about like marbles and coins. But, well, you know what I mean, those questions. And we'll tackle one of those a little bit later. Let's move on to tables. Tables are cool because they offer a visual tool where we can analyze organized data.

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Finally, those tree diagrams. Tree diagrams are graphical representations of possible outcomes in sequential manner. But the thing with tree diagrams is that they're another visual way of organizing possible outcomes. Literally, a tree diagram looks like branches. Okay, that was pretty brief, but I know that you're familiar with tables and diagrams.

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So let's take a better look at counting procedures and how to use them in this next example. Example number one, you have a bag of marbles, three red, six blue, nine green, and five black. What is the probability of selecting one black marble from the bag? To solve this probability problem, you first need to determine the total number of marbles and then find out how many of those marbles are black.

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So finally, you'll divide the number of black marbles by the total number of marbles to get the probability. That sounds like a lot. To find the total number of marbles, you're gonna add up the number of each color. So let's do that. The total was three red, six blue, nine green, and five black, the three plus six plus nine, and then five, that equals 23, 23 marbles.

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Now determine the number of black marbles. Oh, that's right, there's a five. Calculate the probability. Hmm, so I'm gonna divide the number of black marbles by the total number of marbles. The probability of selecting a black marble equals the number of black marbles over the total number of marbles. Wow, actually that was a lot simpler than I thought. That comes out to 5 out of 23. Therefore, the probability of selecting a black marble is 5 out of 23. But how do we write that mathematically?

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Oh, I'm glad you asked. That's a great question. We would express the probability 5 out of 23 in the fraction 5 over 23. Now, think back to the last episode about fractions. We can convert fractions into percentages by dividing the numerator by the denominator, meaning 5 divided by 23.

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So that gives us, let's see, approximately 21.7%. Well, Mercedes, what does that mean? Well, that means you have a 21.7% chance of selecting a black marble out of a bag of 23 marbles. That's pretty cool, huh? Well, I know. But sometimes you're just gonna get those questions that have marbles or coins, like

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in this next example, taken directly from the FTC Mathematics Practice Subtest. You can find this question at fl.nesinc.com. Okay, prepare to calculate. Yes. Example number two, a child has 26 pennies, 15 nickels, 21 dimes, and 18 quarters in a coin bank. When the child picks up the bank, a single coin falls out. What is the probability that that coin is a quarter? Now think back to the last example.

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Where do we start? Oh, we've got to apply the counting procedure, right? That's right. Remember, the counting procedure says we will systematically count the number of possible outcomes. And in a scenario, what is that number of possible outcomes in this scenario? Let's go back to the problem.

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There are 26 pennies, 15 nickels, 21 dimes and 18 quarters. Therefore, the total number of possible outcomes is, oh, let's see, 26 plus 15 plus 21 plus 18. And that equals, let's see, oh, that's 80, 80 possible outcomes. Next, the problem asks, what is the probability that the single coin that fell out is a quarter? Hmm, how many quarters were there? Let me think, 18, there were 18 quarters,

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meaning we have 18 possible outcomes out of 80, the total number of outcomes, possible out of total. Now it's your turn. Answer this. What is the probability of the coin being a quarter? Think about it, but not too long. The probability that the single coin is a quarter is 18 out of 80, right?

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Yes, yes, and no, no. Yes, because it's 18 out of 80, expressed as a fraction 18 over 80, but no, because we can't leave 18 over 80 in that form.

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Why?

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Well, we must simplify the fraction. Fortunately, both numbers are even, so we can simplify them by dividing by two. Um, that gives us 18 divided by two is nine and 80 divided by two is 40. That's nine over 40. Now, you can figure this out using a table or creating a tree diagram too, you know, if you're a visual learner. Let's review.

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Remember that you can use counting procedures, tables and tree diagrams to solve real world probability problems. By understanding how to visualize the number of possible outcomes to the number of total outcomes. Being able to identify and understand when to apply counting procedures

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on the FTCE math subtest will help you solve those probability problems about coins and marbles. And actually factually, maybe it'll help us understand the weather too.

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Yeah.

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For more practice with math concepts, visit ftceseminar.com to study for the test. The important thing is to start studying and start studying today. So you can pass the teacher certification exam. Well, what are you waiting for? Check us out on YouTube at FTC seminar and start studying today. This podcast was recorded at the Pickens multimedia studio at the University of West Florida. This podcast is

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listener supported. Contributions can be made via the listener support link on Spotify. Or, hey, you can buy me a coffee at FTCE seminar dot com. This is your host, Mercedes Musto. This is your host, Mercedes Musto. Join me again on FTCE seminar, a teacher certification podcast.

Transcribed with Cockatoo