If you roll a die (singular of dice), what is the...

If you roll a die (singular of dice), what is the probability you will roll: (a) a $6 ?(\mathrm{b})$ an even number? (c) a number divisible by $3 ?$ (d) If you roll a pair of dice, what is the probability that you will roll two $6 \mathrm{s} ?(\mathrm{e})$ an even number on one and an odd number on the other? (f) matching numbers? (g) two numbers both over $4 ?$

If you roll a die (singular of dice), what is the probability you will roll: (a) a $6 ?(\mathrm{b})$ an even number?
(c) a number divisible by $3 ?$ (d) If you roll a pair of dice, what is the probability that you will roll two $6 \mathrm{s} ?(\mathrm{e})$ an even number on one and an odd number on the other? (f) matching numbers? (g) two numbers both over $4 ?$

Key Concepts

Probability

Probability is the measure of how likely an event is to occur, calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes in a given sample space. This concept is fundamental when dealing with processes like rolling dice where outcomes are equally likely.

Sample Space

The sample space is the set of all possible outcomes of an experiment. In the context of rolling a die or a pair of dice, the sample space contains all distinct outcomes of the roll(s), allowing calculation of event probabilities by counting the outcomes within this space.

Counting Principles

Counting principles, including the rule of product, are used to determine the total number of outcomes in the sample space. When experiments consist of several independent stages (such as rolling multiple dice), the total outcome count is the product of the outcome counts for each individual stage.

Multiplication Rule for Independent Events

The multiplication rule for independent events states that the probability of two independent events occurring together is the product of their individual probabilities. This rule is critical for calculating the probability of combined outcomes in multiple dice rolls, where the result of one die does not affect the other.

Transcript

00:01 In order to answer this question, let's talk about probabilities.

00:03 It's just if you goal a die, a singular of dyes, what is the probability you will go a six? so in this case, remember that a die has six possibilities of getting one, two, three, four, five, and six.

00:20 Okay, so just for a total of six possibilities.

00:23 And what is the probability of getting this number six? well, you have one out of a total of six.

00:30 So you have to divide 1 by 6, and this is going to be equal to 0 .17.

00:36 This is the answer for the first question.

00:38 The next question says, what is the probability of getting an even number? so you have a total of 6 possibilities again.

00:47 And out of 6 possibilities, how many of them are even numbers? you have 1, 2, and 3.

00:54 So you have 3, 6.

00:56 And if you divide this, you're going to get 0 .5.

01:00 So this is the probability of getting an event number.

01:05 The next question says, what is the probability of getting a number divisible by 3? so you have a total of six possibilities and numbers that are divisible by 3 are 3 and 6.

01:19 So you have 1 and 2.

01:21 So you have 2 6 and this is equal to, then by this 2 by 6 is equal to 0 .0 .33.

01:38 The next question says, what is the probability you will, if you got a pair of dice, what is the probability that you will go to six? so if you get, if you got a pair of dice, what is the probability that you will go to six? so it means that you want first one six and improbabilities end means that you have to multiply.

02:00 You want also another six.

02:01 So you have another one six.

02:04 So you have to multiply this and you're going to get one 36.

02:07 Let me divide this to see the answer in decimal.

02:11 And this is going to be equal to 0 .028.

02:19 Okay? now let's go for the next question.

02:23 It says the probability an even number on one and an odd number on the other.

02:29 So you have a total of six possibilities and you want first an even number.

02:33 So you have one, two, and three...