Algebra I Math Question

[Reefgazer]

This is a probability problem in DD's Algebra I book:

 

5 +6 +5 +4 +3+2+1

_______________.

36

 

 

DD and I don't understand why the top numbers are there and how they got 26. I mean I know the numbers together add up to 26, but I don't understand why those numbers were chosen to be in the numerator.

[SparklyUnicorn]

Hm...

 

So it's those numbers on top with 36 on the bottom?

 

 

[Reefgazer]

Yes. I don't understand why.

 

Hm...

 

So it's those numbers on top with 36 on the bottom?

[8filltheheart]

Without any context, it is hard to understand what the problem is demonstrating/asking.  (BTW, do you have a typo?  Is the denominator be 26 or 36?) 

[Reefgazer]

Oh, duh...So tired today I didn't actually post the problem. Anyway, the problem reads: Margaret rolls two fair dice. what is the probability that the sum of the numbers rolled is not more than eight? The denominators 36; the numerator is 26.

 

quote name="8FillTheHeart" post="6931341" timestamp="1459979907"]

 

Without any context, it is hard to understand what the problem is demonstrating/asking. (BTW, do you have a typo? Is the denominator be 26 or 36?)

[Cosmos]

Oh, duh...So tired today I didn't actually post the problem. Anyway, the problem reads: Margaret rolls two fair dice. what is the probability that the sum of the numbers rolled is not more than eight? The denominators 36; the numerator is 26.

 

When all possibilities are equally likely, you compute probability by dividing the number of possibilities of the desired result by the total number of possibilities.

 

If you roll two dice, there are 36 total possible outcomes:

1 1

1 2

1 3

1 4

1 5

1 6

2 1

2 2

.

.

.

and so on all the way to

6 4

6 5

6 6

 

Do you see why that makes 36 total possible outcomes?

 

Now you need to find the number of outcomes that give the desired result, in this case that the sum is not more than 8. You need to find all of those possibilities. Here they are:

 

1 1, 1 2, 1 3, 1 4, 1 5, 1 6

2 1, 2 2, 2 3, 2 4, 2 5, 2 6

3 1, 3 2, 3 3, 3 4, 3 5

4 1, 4 2, 4 3, 4 4

5 1, 5 2, 5 3

6 1, 6 2

 

The other ones I didn't list like 5 4 sum to more than 8.

 

If you add those up, you'll see that there are 26. So the answer would be 26/36.

 

Now they found their answer a slightly different way. I suspect that they grouped theirs this way:

 

Dice that add to 1: no possibilities

Dice that add to 2: 1 1 = 1 possibility

Dice that add to 3: 1 2, 2 1 = 2 possibilities

Dice that add to 4: 1 3, 2 2, 3 1 = 3 possibilities

Dice that add to 5: 1 4, 2 3, 3 2, 4 1 = 4 possibilities

Dice that add to 6: 1 5, 2 4, 3 3, 4 2, 5 1 = 5 possibilities

Dice that add to 7: 1 6, 2 5, 3 4, 4 3, 5 2, 6 1 = 6 possibilities

Dice that add to 8: 2 6, 3 5, 4 4, 5 3, 6 2 = 5 possibilities

 

Adding them up that way you get 1 + 2 + 3 + 4 + 5 + 6 + 5, which of course is also 26.

 

Edited April 6, 2016 by Cosmos

[regentrude]

Probability is number of desired outcomes divided by number of all possible outcomes

Possibilities to achieve goals:

sum is 2: occurs once if dice show 1 and 1

sum is 3: occurs twice if dice show 1 and 2 or 2 and 1

sum is 4: (1,3), (3,1), (2,2) three possibilities

sum is 5: (1,4) (4,1) (2,3) (3,2) four

sum is 6: (1,5) (5,1) (2,4) (4,2) (3,3) five

sum is 7: six possibilities

sum is 8: (2,6), (6,2), (3,5) (5,3), (4,4) five

so total "good outcomes" are 5+6+5+4+3+2+1

divide by number of all possible outcomes which is 36

 

ETA: A cleverer way to solve the problem would be to find the probability that the sum is NOT 8 or less because this is quicker since the possible sums are only 9,10,11 and 12, and subtract that from 1.

Edited April 6, 2016 by regentrude

[Reefgazer]

Thanks, all! DD and I were adding wrong and couldn't figure out why our simple wrong addition wasn't giving us the correct answer. 😂