Selecting Marbles A bag contains 9 red marbles, 8 white marbles, and 6 blue marbles. Randomly ch

Selecting Marbles A bag contains 9 red marbles, 8 white...

Key Concepts

Addition Rule for Mutually Exclusive Events

This rule is used when calculating the probability of one event or another occurring, particularly in cases where outcomes cannot happen simultaneously. When computing the probability of events that are mutually exclusive, you simply add their individual probabilities. This is especially useful in scenarios where multiple disjoint outcomes lead to a desired result.

Random Experiment and Sample Space

This concept involves considering all possible outcomes when conducting an experiment, such as drawing marbles from a bag. Understanding the sample space — the set of all outcomes — is fundamental in probability theory, as it provides the basis for calculating the likelihood of specific events occurring.

Without Replacement

When items are drawn without replacement, it means that each item drawn is not returned to the original set. This makes subsequent draws dependent events because the composition of the set changes after each draw, affecting the probabilities of later events.

Dependent Events and Conditional Probability

In scenarios where draws are made without replacement, each event affects the probability of the next event occurring. This is known as dependency. Conditional probability is used to describe the probability of an event given that another event has already occurred, enabling the adjustment of probability calculations based on changes in the sample space.

Multiplication Rule for Dependent Events

This rule states that the probability of two dependent events both occurring is the product of the probability of the first event and the probability of the second event given that the first has occurred. It is crucial when dealing with sequential events where the outcome of the first event influences the outcome of the second.

Transcript

00:01 Okay, in this problem, there's a lot going on, but i have drawn a tree here to help break it down.

00:06 So first, i'm going to explain that.

00:09 First of all, we know that we are given nine red marbles, eight white marbles, and six blue marbles, giving us a total of 23.

00:17 The tree, if you've never seen it before, each stage of the tree represents a different set of choices.

00:24 So stage one of our tree is going to be our first pick of a random marble.

00:28 And stage two is going to represent the second pick of the marble.

00:32 And also, we want to keep in mind that we are not replacing the marbles when we are first picking them.

00:38 So after we pick a first one here, there are only 22 remaining, hence the 22 in all the denominators here.

00:46 Another important note about probability trees is that the probability sitting here is the chance of pulling a red on the first pick, for instance.

00:56 But when you move over to the second stage here, these branches represent conditional probabilities.

01:04 For example, right here, this 8 over 22, that is the probability of picking a white marble for the second pick given that you just picked a red marble.

01:18 Further, just to break down our understandings here of why these numbers are where they are, i would believe that the 9 over 23, the 8, and a 6 over 23 are all.

01:29 Pretty standard knowledge because that is just part over the total for these probabilities.

01:36 But for the second pick, this stage is assuming that you've picked red.

01:40 So first of all, you have 22 left, but now you can see that you've won less red.

01:44 So your probability of picking red is now 8 instead of 9, 8 over 22 because you also have one less marble.

01:50 The same happens here with our whites.

01:52 We now have one less.

01:54 And with the blues, we have one less.

01:57 Last two things about probability trees is that each set of branches, like for instance, what i just circled here or here, you will notice that all of these probabilities, these three add to one.

02:10 And these three add to one, and these three add to one, and even these add to one.

02:17 And our last little bit about probability trees is that the paths of the trees, the path that you take, like this, for instance, these paths will multiply to find your intersection probability, which would be the probability of both occurring.

02:31 For instance, drawing a red on the first pick and a red on the second pick, you would multiply the 9 over 23 times your 8 over 22 to get this value.

02:39 So now using all of this information, also i just want to note that, you know, fractions can be a pain to plug into your calculator.

02:48 So i have gone ahead and done these calculations for you ahead of time so you can plug in the decimal values.

02:54 But i did think it was easier to understand conceptually what was going on with the fractions in our probability tree.

03:01 So now let's attack part a.

03:03 Part a wants us to find the probability that the first marble is red and the same.

03:09 Second is white.

03:11 So to do this, this is the power of our probability tree here...

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