Two number cubes are rolled. Find the probability of each outcome. Both numbers rolled are even.

Name: Problem 38, Chapter 4: Geometry
Uploaded: 2020-03-28T05:32:46-08:00
Duration: 6 min 39 s
Channel: Jason Taylor-Pestell
Description: Two number cubes are rolled. Find the probability of each outcome. Both numbers rolled are even.

Two number cubes are rolled. Find the probability of each...

Key Concepts

Classical Probability

This concept defines probability as the ratio of the number of favorable outcomes to the total number of possible outcomes. It is used in determining the likelihood of an event occurring in scenarios where every outcome is equally likely.

Sample Space

The sample space encompasses all possible outcomes of an experiment. In the context of rolling two cubes, it consists of every possible combination of numbers that can appear, which is essential for determining both the total outcomes and the events of interest.

Independent Events

Independent events are those events where the outcome of one event does not influence the outcome of another. When rolling two cubes, the outcome on one cube does not affect the outcome on the other, allowing the probabilities to be multiplied.

Counting Techniques

Counting techniques refer to the methods used to calculate the number of favorable outcomes. These methods often involve enumeration or application of rules like the multiplication principle, particularly when considering events such as obtaining even numbers on each cube.

Transcript

00:01 All right.

00:01 So this problem and the next few problems are what are considered spiral review.

00:07 So there are parts of things that you have learned, hopefully, in a previous course or in a previous lesson in this book.

00:14 So this one is from a previous course.

00:18 Hopefully you have seen some probability.

00:22 But if you haven't, that's okay.

00:23 This is a relatively simple concept.

00:29 We just need to figure out.

00:31 What the types of events are.

00:35 So in probability with number cubes, a number cube looks like this.

00:41 You may see them with dots as well, and you may have played with them in different types of games.

00:47 It's a chance object.

00:50 It provides some type of possible outcomes, and each of them is technically equally likely as the others.

01:00 And so in a typical number cube, there are six sides, one, two, three, four, five, and six.

01:08 And a very typical question will say, what will happen when you roll two of these? what's the likelihood or the probability that something is going to happen? and so whenever that's the case, the first thing you need to do is identify what are the possibilities of what can happen.

01:28 Not what you want to happen, but what can happen.

01:31 Here i've shown a strategy that i've learned for when you roll two different number cubes.

01:39 So i've notated the first one being a white number cube, the second one being black, and that's just to keep track of one of them and the second one.

01:48 You could also just say this is the first and the second one.

01:51 This then lists all the possibilities for one of them.

01:54 So, for example, the probability of rolling a two on the white number cube is one time out of six possibilities.

02:06 The probability of rolling a prime number on the die would be one, two is a prime number, three is a prime number, five is a prime number.

02:20 One is the identity, so it is not a prime number.

02:23 So i counted three of them out of the six of them.

02:28 So the probability that you would roll a prime number, just rolling one die, is three out of six, or one half, or 50%.

02:38 So that's the basic idea of probability.

02:41 So in this one, we need to find the probability that if you were to roll two number cubes and look at the result, if they are both even numbers, then that's what we want to happen.

02:52 And so what we want to happen goes here in the numerator of our fraction.

02:56 The total is what goes down below.

03:00 And the total is found when you create whatever strategy is helpful for making sure you identify all the possibilities.

03:07 So the first possibility is that the white number cube is a one.

03:11 If the white number of cube is a one, the black one could be a one, a two, a three, a four, five, or six.

03:18 Same thing for if the white one was a two.

03:21 The black one could be a one, two, three, four, five, or six.

03:25 And it goes on.

03:26 Each number that the white rolls, the black one is completely independent.

03:32 It's not dependent on what the white dye comes out to.

03:37 And so their two results are combined in this way.

03:39 So instead of having only six possibilities, we have six rows...