Use the following spinners to construct a probability model for each experiment. (Image can't co

Use the following spinners to construct a probability model...

Key Concepts

Probability Model

A probability model is a structured representation of a random experiment, which includes defining the sample space (all possible outcomes) and assigning probabilities to these outcomes. This model helps in visualizing and calculating the likelihood of various events occurring during an experiment.

Sample Space

The sample space is the complete set of all possible outcomes that can occur in an experiment. For experiments involving multiple steps or stages, the sample space is often constructed by considering the cross-product of the outcomes from each individual stage.

Independent Events

Independent events are events whose outcomes do not influence one another. In experiments with multiple stages, such as sequential spins of spinners, the outcome of one spin does not affect the outcomes of the subsequent spins, allowing for simpler probability calculations.

Multiplication Rule

The multiplication rule is used to determine the probability of a sequence of independent events occurring together. It states that the probability of the combined event is the product of the probabilities of each individual event, which is fundamental in solving multi-step probability problems.

Compound Events

Compound events involve the occurrence of two or more simple events together, often using conditions such as 'or', 'and', or specific sequences. Assessing compound events requires careful consideration of whether the events are mutually exclusive or overlapping and applying the appropriate rules to calculate their probability.

Transcript

00:01 We're going to construct a probability model for spinner three followed by spinner one twice.

00:05 So it's going to be spinner three, spinner one, spinner one.

00:08 Then we're going to get the probability of being an f followed by a one or a three, and then followed by a two or a four.

00:15 If we want to construct this probability model, a list is a great way, but a tree always does give us some type of organization to get started.

00:22 So if we're going to start by looking at a tree, why don't we start with our spinner three, which is either, forward or backwards.

00:35 Now, since we started with your forward or backward, we would next go to spinner 1.

00:39 If we start with f followed by spinner 1, our next possibility is either be 1, 2, 3, or 4.

00:45 So we have f and 1, f2, f3, or f4.

00:53 Now, from this point, we'd go to spinner 1 again.

00:57 So if we had f and then a 1 first, we would still have four possibilities for what comes next, either 1, 2, 3, or 4.

01:04 So we have f11, f12, f13, and f14.

01:11 The same thing could be said if you started with f and 2, it could then either go 1, 2, 3, or 4.

01:18 Whop, 2, 3, or 4.

01:20 So it could be f21, f22, f2, or f24.

01:28 And we can keep this pattern going now that we've established it.

01:32 The next thing would to say, hey, what if we start with f and then three? well, we could have f31, f3 and then a two, f3 and then a three, or f3 and a four.

01:44 Or if we started f then four, we could follow it up with a one.

01:47 We could go f and four and then two, f43, f4, or f4.

01:54 As you can see, we've done all the possibilities that started with a forward on spin three.

02:01 We do have 16 things.

02:03 So we should have 30.

02:04 We should have 30.

02:04 32 when we get done.

02:06 One thing that i didn't do at the beginning, it's always a great idea, is how many things should we have? well, there's two possibilities on spinner three, followed by four on spinner one, followed by four on spinner one the other time, which gives us eight times four, 32 possibilities...

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