Use the rules of probability to calculate these...

Use the rules of probability to calculate these probabilities: 1. What is the probability of rolling a five on one throw of a fair six-sided die? 2. What is the probability of not rolling a five on one throw of a fair six-sided die? 3. If you roll a five on your first throw of a fair six-sided die, what is probability of rolling another five on a second throw of that die? 4. If you roll two fair six-sided dice one time, what are the chances that both of the dice will come up a five? 5. If you roll two fair six-sided dice one time, what are the chances that one or the other (or both) of the dice will come up a five? 6. If you roll two fair six-sided dice one time, what are the chances that one and only one of the dice will come up a five? 7. If you roll two fair six-sided dice one time, what are the chances that at least one of the dice will come up a five? 8. If you roll two fair six-sided dice one time, what are the chances that at least one of the dice will not come up a five? 9. If you roll six fair six-sided dice one time, what are the chances that at least one of the dice will come up a five? 10. If you roll six fair six-sided dice one time, what are the chances that at least one of the dice will not come up a five?

Use the rules of probability to calculate these probabilities:
1. What is the probability of rolling a five on one throw of a fair six-sided die?
2. What is the probability of not rolling a five on one throw of a fair six-sided die?
3. If you roll a five on your first throw of a fair six-sided die, what is probability of rolling another five on a second throw of that die?
4. If you roll two fair six-sided dice one time, what are the chances that both of the dice will come up a five?
5. If you roll two fair six-sided dice one time, what are the chances that one or the other (or both) of the dice will come up a five?
6. If you roll two fair six-sided dice one time, what are the chances that one and only one of the dice will come up a five?
7. If you roll two fair six-sided dice one time, what are the chances that at least one of the dice will come up a five?
8. If you roll two fair six-sided dice one time, what are the chances that at least one of the dice will not come up a five?
9. If you roll six fair six-sided dice one time, what are the chances that at least one of the dice will come up a five?
10. If you roll six fair six-sided dice one time, what are the chances that at least one of the dice will not come up a five?

Key Concepts

Addition Rule

The addition rule is used to calculate the probability that at least one of several mutually exclusive events occurs. In scenarios where events cannot occur simultaneously, it simplifies determining the overall probability by summing up individual probabilities, though care must be taken with events that might not be mutually exclusive.

Multiplication Rule

The multiplication rule is used for finding the probability of two or more independent events occurring together. The probability of all events happening is the product of their individual probabilities. This rule is particularly useful when calculating the likelihood of specific combinations in multi-dice rolls.

Complement Rule

The complement rule is a fundamental probability concept where the probability of an event occurring plus the probability of the event not occurring equals one. This rule allows us to compute the probability of an event by subtracting the probability of the event's complement from one, which is particularly useful for 'at least one' scenarios.

Sample Space

The sample space of an experiment is the set of all possible outcomes. For a six-sided die, the sample space is {1, 2, 3, 4, 5, 6}. Understanding the sample space is critical for determining probabilities because it provides the denominator when calculating the probability of one or more specific events.

Uniform Probability

This concept refers to situations where every individual outcome has an equal chance of occurring. In the context of rolling fair dice, it means that each face of the die (for example, numbers one through six) has the same probability since the die is not biased towards any particular number.

Independent Events

This concept denotes that the outcome of one event does not affect the outcome of another. For instance, successive rolls of a fair die are independent; the result of one roll doesn’t influence the result of the next roll, which is essential when calculating joint probabilities using the multiplication rule.

Transcript

00:01 In this problem, we're rolling three six -sided dye, and part a is asking us to find the probability that we get triples.

00:08 So triples means it could be three -ones or three -twos or three -threes.

00:13 So suppose we have three -ones, that would be one -six times one -six times one -six, or it could be three - twos, in which case it would be the same thing, one -six, one -six, or three -threes and so on.

00:29 So actually we're going to get six different options here.

00:35 The ones, two, three, fours, fives, and sixes.

00:38 So this will end up being one over 36.

00:43 In part b, this is a conditional probability.

00:48 So first of all, we are only going to consider all the options that have a sum less than are equal to eight, and we're going to have to generate that in a second.

00:57 The numerator is considering the number of triples.

01:00 So think about the number of triples that would be less than or equal to 8.

01:04 Well, 1 -1 -21 -22.

01:07 333 is already a sum of 9.

01:09 So there's only two options of triples for us to consider, but we've got to figure how many options we're considering from.

01:17 So we actually have to map this out, and we're going to do this on the bottom right here.

01:23 Let's list sample space.

01:25 Some less than or equal to 8.

01:26 Well, we could have 111, 112, 113, 114, 115, 116.

01:36 And i'm going to recognize here that these guys all have three different arrangements.

01:45 Could be, let's take 116, for example, that could be 161 or 611.

01:51 Those are all different combinations, or different permutations.

01:54 So we would have three times this many number of combinations with these sets of numbers.

02:02 But in addition to that, we could also have starting with a 1 and then 2 -2.

02:10 1 -2 -3, 1 -2 -4.

02:14 That would add up to 7.

02:16 1 -2 -5.

02:19 And let's try 1 -3 -13 -1 -3.

02:23 1 -3 -4 is okay as well.

02:27 Because that would add up to eight and we can do one four one which we've already counted because it's one one four one four two is counted one four two is counted one four three we've done already and one four would exceed that so this is it that's the all the combinations that start with you know that have a one in them so this one will have three combinations these ones are three distinct objects so there will be six combinations is it three factorial.

03:00 This will have three combinations and three distinct objects here will have six combinations.

03:07 Now let's start with two.

03:09 Two, two, two, two, two, two, two, three, two, four.

03:16 We can't go to two, five, because that would exceed.

03:19 We could have two, three, one, which if we've done already, 232, which is the same as 223.

03:35 So 233, we have not done.

03:42 And we can't go to 234.

03:44 We can go 241, which we've done.

03:49 242 is done.

03:52 243 would already exceed.

03:54 So this is it.

03:56 So 223, 224...