Question

If two dice are rolled, what is the probability that the uppermost faces do not sum to five or are not a product of 4?

Answer 1

List all possibilities where the numbers on the dice add up to 5:

Answer 2

There are 3

Answer 3

Can you list them?

Answer 4

1,4 2,3 3,2 4,1

Answer 5

Isn't that 4 possibilities?

Answer 6

oh yes sorry

Answer 7

List all possibilities where the sum is a product of 4. That is, the sum is 4, 8 or 12.

Answer 8

1,4 2,2 4,1

Answer 9

Sum is 4: (1,3); (2,2); (3,1)
Sum is 8: (2,6); (3,5); (4,4); (5,3); (6,2)
Sum is 12: (6,6)

Earlier,
Sum is 5: (1,4); (2,3); (3, 2); (4,1)

Go through the entire list and see if there are any duplicates.
The count the total number of possibilities without duplicates.

Answer 10

Why is product of 4 equal to sum of 4 8 and 12?

Answer 11

okay theres no duplicates

Answer 12

The question is not worded properly and can be interpreted in more than one way.
"the uppermost faces do not sum to five or are not a product of 4?"
The way I interpreted it is: products of 4 are: 4x1=4; 4x2=8; 4x3=12
the sum should not be a product of 4. That is, the sum should not be 4, 8 or 12.

Answer 13

How?

Answer 14

Apparently what the product one means is that die1 + die2 = 4

Answer 15

I meant die 1 x die 2 = 4

Answer 16

Yes, that is what I mean by multiple interpretation.
Should the sum not be a product of 4? or
Should the product not be 4?

Answer 17

If you are sure they mean die 1 x die 2 = 4,
then the possibilities are: (1,4); (2,2); (4,1)

Answer 18

yes i am sure they mean that

Answer 19

Sum is 5: (1,4); (2,3); (3, 2); (4,1)
Product is 4: (1,4); (2,2); (4,1)

How many distinct possibilities are there? Be careful of counting duplicates.

Answer 20

Count ALL possibilities. Both sum is 5 and product is 4. (without duplicates)

Answer 21

without counting the same possibility twice.

Answer 22

There are 2 that satisfy both and those are (1,4) and (4,1)

Answer 23

So don't count them a second time. How many total possibilities?

Answer 24

5

Answer 25

Yes. And if we roll a pair of dice how may total possible outcomes?

Answer 26

36

Answer 27

Yes. And the question asks: "DO NOT sum to five or are not a product of 4?"
We found 5 possibilities that we don't want. So we are left with 36 - 5 = 31 desired outcomes. So the probability is: 31/36.