involving the sum of the numbers showing on two fair dice.
(1) What is the probability that exactly one die shows a 6 given that the sum of the numbers is 9?
(2) What is the probability that the sum of the numbers is 9 given that exactly one die shows a 6?
(3) What is the probability that the sum of the numbers is 9 given that at least one die shows a 6?

- Pulsified333

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- Pulsified333

@jim_thompson5910

- jim_thompson5910

which one are you stuck on? how far did you get?

- Pulsified333

i haven't gotten any yet. lets start with #1 @jim_thompson5910

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## More answers

- jim_thompson5910

it tells us in #1, that `given that the sum of the numbers is 9`
so we know, for a fact, that the two dice add to 9

- jim_thompson5910

what are the ways to add to 9 with 2 dice?

- Pulsified333

4&5

- jim_thompson5910

yep, what else? can you list ALL of the ways to add to 9?

- Pulsified333

3&6, 4&5

- jim_thompson5910

so put that all together we have
3+6
4+5
5+4
6+3
there are only 4 ways to do this

- jim_thompson5910

of these 4 ways, how many have exactly one "6" in them?

- jim_thompson5910

you sure?

- Pulsified333

2?

- jim_thompson5910

yeah 2
3+6 and 6+3

- jim_thompson5910

it would be a fraction
Probability of exactly one 6, given sum of 9 = (# of sums with 6 in them)/(# of ways to add to 9) = 2/4 = 1/2

- jim_thompson5910

` given that exactly one die shows a 6`
so we know die A is 6 or die B is 6 (both cannot be 6 at the same time)

- jim_thompson5910

if die A is 6, then what are the possibilities for die B?

- Pulsified333

5 and 4?

- jim_thompson5910

B could also be 1,2,3
basically it could be anything but 6

- jim_thompson5910

if die A is 6, then die B could be 1,2,3,4,5
similarly
if die B is 6, then die A could be 1,2,3,4,5

- Pulsified333

but doesn't the sum have to be 9?

- jim_thompson5910

no, the given here is that one die is 6. That's it

- jim_thompson5910

this dice chart may help

##### 1 Attachment

- Pulsified333

ok so die B could be 1,2,3,4,5

- jim_thompson5910

how many possible outcomes are there if exactly one die is 6?

- Pulsified333

but it says What is the probability that the sum of the numbers is 9

- jim_thompson5910

we'll get there

- jim_thompson5910

in the chart I posted, mark the row that has 6 and the column that has 6. But don't mark the cell that has both 6's. How many cells did you highlight?

- Pulsified333

5?

- jim_thompson5910

you should count 5 along the '6' row
and 5 along the '6' column
so 10 in total

- jim_thompson5910

how many of those 10, have a sum of 9?

- Pulsified333

2?

- jim_thompson5910

yes, 6+3 and 3+6
we have 2 sums that have a sum of 9
out of 10 outcomes
so 2/10 = 1/5 is the answer to #2

- Pulsified333

oh ok, thanks

- jim_thompson5910

and for #3, it's almost identical to #2
BUT
the key word in #3 is `at least`
so it's possible to have both dice be 6

- Pulsified333

6&3 3&6

- Pulsified333

so is it 1/5? @jim_thompson5910

- jim_thompson5910

we have 2 ways to add to 9 (6+3 or 3+6)
however, instead of 10 outcomes, we actually have 11. Again, it's possible to have both dice be 6
so it's actually 2/11

- Pulsified333

thank you :)

- jim_thompson5910

no problem

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