# Thread: Probability of events (rolling dice)

1. ## Probability of events (rolling dice)

The question can be found here:

Please don't provide an explanation. Just give me a hint.

A 6-sided number cube, with faces numbered 1 through 6, is to be rolled twice. What is the probability that the number that comes up on the first roll will be less than the number that comes up on the second roll?

a. 1/4
b. 1/3
c. 5/12
d. 7/12
e. 1/2

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I have thought about this question for a while now. It doesn't make sense to me because I'm not given any information about the outcomes. Can I have a hint?

2. This is how that can happen.
$\displaystyle \begin{array}{*{20}c} {(1,2)} & {(1,3)} & {(1,4)} & {(1,5)} & {(1,6)} \\ {} & {(2,3)} & {(2,4)} & {(2,5)} & {(2,6)} \\ {} & {} & {(3,4)} & {(3,5)} & {(3,6)} \\ {} & {} & {} & {(4,5)} & {(4,6)} \\ {} & {} & {} & {} & {(5,6)} \\ \end{array}$

3. "The number on the first roll is less than the number on the second roll" is the same as "the number on the second roll is larger than the number on the first roll"

If A and B are independent events than P(A and B)= P(A)P(B). If A and B are mutually exclusive events then P(A or B)= P(A)+ P(B). (If you are given problems like this to solve, you are expected to know those.)

Let "P(x= a)" be the probability that the number rolled is a. Let "P(x> a)" be the probability that the number rolled is larger than a.

If a "1" were rolled on the first roll, then the probability that the number on the second roll is larger is P(x> 1)= 5/6 (any number except 1). Of course, theprobabilty that the first number rolled is 1 is P(1)= 1/6 so the probabilty of that happening is P(1)*P(x> 1)= (1/6)(5/6)= 5/36.

If a "2" were rolled on the first roll, then the probability that the number on the second roll is larger is P(x> 2)= 4/6 (any number except 1 or 2). The probability of that happening is P(2)*P(x> 2)= (1/6)(4/6)= 4/36.

etc.

The probability that the second number is larger than the first number is
P(x=1)P(x> 1)+ P(x=2)P(x>2)+ P(x= 3)P(x> 3)+ P(x= 4)P(x> 4)+ P(x= 5)P(x> 5)+ P(x= 6)P(x> 6).

4. The notation is foreign to me but I understand your explanation. Can you direct me to a reference (either book or online article) where I can find an explanation for probability notation?

Also, is there a reference which teach me various topics in probability? Usually, when I study probability in math, only permutations and combinations are the only topics that are covered.

5. Thanks! I understand now that it comes to: 15/36; 5/12.

What does the "36" represent? The total number of roles to attain all of those results?

6. Originally Posted by Masterthief1324
Thanks! I understand now that it comes to: 15/36; 5/12.

What does the "36" represent? The total number of roles to attain all of those results?
$\displaystyle 36 = 6^2$ is the number of ways to roll two 6-sided dice when order is considered. That is, (1,2) is considered distinct from (2,1). Alternatively, you can think of it in terms of the dice being coloured or otherwise identifiable; for example, if we have a green and a red die, then rolling a 1 on the green die and a 2 on the red die is considered different from rolling a 2 on the green die and a 1 on the red die.