Solve each probability problem. Rolling a Die Twice If a single die is rolled twice, then what i

Solve each probability problem. Rolling a Die Twice If a...

Key Concepts

Complement Rule

The complement rule states that the probability of an event not occurring is one minus the probability of the event occurring. This approach is especially helpful when it is easier to calculate the probability of the event’s complement and then subtract from one.

Multiplication Rule

The multiplication rule is used to find the probability that a sequence of independent events happens by multiplying the probability of the first event by the probability of the second event, and so on. This rule simplifies calculating combined probabilities in multi-step experiments.

Addition Rule

The addition rule is used for finding the probability that at least one of several mutually exclusive events occurs by adding their individual probabilities. It becomes particularly useful when an event can occur via multiple distinct and non-overlapping ways.

Independent Events

Independent events are those where the outcome of one event does not affect the outcome of another. Recognizing independence allows for the use of the multiplication rule, as the probability of a sequence of independent events occurring is the product of their individual probabilities.

Sample Space

The sample space is the complete set of all possible outcomes of a random experiment. In problems involving two trials or stages, the sample space provides the framework for computing probabilities by enumerating every possible sequence of outcomes and assigning them equal likelihood.

Fundamental Counting Principle

The fundamental counting principle states that if one event can occur in m ways and another independent event can occur in n ways, then the two events together can occur in m × n ways. This principle is essential for determining the total number of outcomes in multi-step experiments.

Transcript

00:02 If a single die is rolled twice, then what is the probability of getting a 1 followed by a 6? so the probability of getting a 1, then 6.

00:21 And this only happened once, so it's 1 out of 36.

00:27 So that's a.

00:28 Then case b a sum of four.

00:37 So we want to find probability that the numbers give us a sum of four.

00:45 Now, a sum of four, let's see what are the possibilities.

00:50 It's either it's two and two or three and one and one and three.

00:58 What is the other? nothing else so there are only three cases so it's three out of 36 which is simplified as one out of 12 okay then i'll see a 5 on the second role all right so what is the probability that you get a 5 regardless of what you got in the first case so in the second and roll, you get a 5.

01:38 So it could be a 1 and a 5, a 2 and a 5, a 3 and a 5, a 4 and a 5.

01:49 A 5 and a 5, 6 and a 5.

01:52 So these are the only cases that are available where you get a 5 in the second case.

02:03 I can't think of another.

02:05 So it's 1, 2, 3, 4, 5, 6.

02:09 So it's 6 out of 36, which is 1 out of 6.

02:18 Now, the next person says, no more than 2 4s.

02:28 No more than 2 4s.

02:35 Okay? so in this case you just have to find, when they say no more than two force, it means what we don't want are two force.

02:59 So we can have one four or no four at all.

03:08 Or we can have four.

03:12 So it means if we can have two fours, it's okay.