Question

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?

Answer 1

@jim_thompson5910

Answer 2

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

Answer 3

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

Answer 4

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

Answer 5

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

Answer 6

4&5

Answer 7

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

Answer 8

3&6, 4&5

Answer 9

so put that all together we have

3+6
4+5
5+4
6+3

there are only 4 ways to do this

Answer 10

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

Answer 11

you sure?

Answer 12

2?

Answer 13

yeah 2

3+6 and 6+3

Answer 14

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

Answer 15

` 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)

Answer 16

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

Answer 17

5 and 4?

Answer 18

B could also be 1,2,3

basically it could be anything but 6

Answer 19

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

Answer 20

but doesn't the sum have to be 9?

Answer 21

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

Answer 22

this dice chart may help

Answer 23

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

Answer 24

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

Answer 25

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

Answer 26

we'll get there

Answer 27

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?

Answer 28

5?

Answer 29

you should count 5 along the '6' row
and 5 along the '6' column

so 10 in total

Answer 30

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

Answer 31

2?

Answer 32

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

Answer 33

oh ok, thanks

Answer 34

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

Answer 35

6&3 3&6

Answer 36

so is it 1/5? @jim_thompson5910

Answer 37

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

Answer 38

thank you :)

Answer 39

no problem