# Are these two dice rolls dependent or independent?

In summary, the two events, A and B, are independent if the probability of both events happening is the same no matter what the first event is.
Summary:: Roll a dice twice.

Event A: First dice roll yield 2 or 5.
Event B: Sum of the two results are atleast 7

Is A and B independent?

If they are independent then
$$P(A \cap B) = P(A) * P(B)$$

P(A) = 2/6 = 1/3
P(B) = 1 - 9/36 = 27/36
$$P(A \cap B) = 7/36 \neq P(A)*P(B) = 1/4$$Yet book says they are independent.

##P(B)## should just be a case of counting the number of outcomes that sum to greater or equal than 7, out of the possible 36 outcomes.

P(B) = 1 - 9/36 = 27/36
Yet book says they are independent.
That P(B) is definitely not right.

One way to think about independence is whether the first event influences the second. In this case it appears that the probability of the second event depends on the first. But, does it? These two events look like they should be dependent, but are they really?

If you had to bet at the start on getting a total of ##7## or more, what would the probability be? And, if you were told that the first die was either a 2 or a 5, what would the revised probability be?

To put this formally, two events are independent if:
$$P(B|A) = P(B)$$
This says that ##B## is just as likely whether or not ##A## happens. Hence, alternatively:
$$P(A \cap B) = P(A)P(B|A) = P(A)P(B)$$

Last edited:
Hm, yea I calculated it to be 9 cases but there are 12 giving an outcome below 7.
P(B) =
1,5
1,4
1,3
1,2
1,1
2,1
2,2
2,3
2,4
3,1
3,2
3,3

P(B) = 1 - 12/36 = 24/36 = 2/3
P(A) * P(B) = 2/9

P(A & B) =
2,5
2,6
5,2
5,3
5,4
5,5
5,6
= 7/36

Would any of these have one less or one more combination then P(A&B) = P(A)P(B)
But as it is, there isnt.

Hm, yea I calculated it to be 9 cases but there are 12 giving an outcome below 7.
P(B) =
1,5
1,4
1,3
1,2
1,1
2,1
2,2
2,3
2,4
3,1
3,2
3,3

You forgot:

4,1
4,2
5,1

P(B)=21/36

I get that they are independent also.

## 1. Are the two dice rolls dependent or independent?

The answer to this question depends on the context in which the dice rolls are occurring. If the rolls are being done simultaneously, then they are considered independent. However, if the first roll affects the outcome of the second roll (e.g. rolling doubles in a game of Monopoly), then they are considered dependent.

## 2. How do you determine if two dice rolls are dependent or independent?

To determine if two dice rolls are dependent or independent, you need to look at the relationship between the two rolls. If one roll affects the outcome of the other, then they are dependent. If the rolls are completely unrelated, then they are independent.

## 3. Can two dice rolls be both dependent and independent?

No, two dice rolls cannot be both dependent and independent at the same time. They are either one or the other, depending on the context in which they are being used.

## 4. What is an example of dependent dice rolls?

An example of dependent dice rolls would be rolling two dice in a game of Yahtzee. The first roll affects the outcome of the second roll, as players can choose which dice to keep and which to re-roll.

## 5. Are there any real-life situations where two dice rolls would be considered dependent?

Yes, there are real-life situations where two dice rolls would be considered dependent. For example, if you are rolling two dice to determine the number of spaces you move in a board game, the first roll will affect the outcome of the second roll. Another example would be rolling two dice to determine the outcome of a game of craps, where the first roll affects the odds of the second roll.

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