Two fair dice (one red and one green) are rolled. What is the probability that the sum is 5, given that the green one is either 4 or 3?

Let #A# be the event that the sum of the two dice is #5#
Let #B# be the event that the green die is either a #3# or #4#

Then we want #P( A | B)# which we calculate using the conditional probability formula:

# P( A | B) = (P(A nn B)) / (P(B)) #

Consider first #P(A nn B)# which we can calculate using

# P(A nn B) = (n(A nn B)) / (n(T)) #

Where, #n(T)# is the total number of possible outcomes. As there are two dices then each has #6# possible outcomes, making #n(T)=36#.

And, #n(A nn B)# is the number of outcomes where the total is #5# and the green die is a #3# or a #4#.

If #G=3 => R=2 #
If #G=4 => R=1 #

And so #n(A nn B)=2#, and therefore we have:

# P(A nn B) = 2/36 = 1/18 #

Now, let use calculate #P(B)# in a similar fashion:

# P(B) = (n(B))/(n(T)) #

Where #n(B)# is the number of outcomes for which the green die is a #3# or a #4#. For this outcome we have:

If #G=3 => R=1,2,3,4,5,6 #
If #G=4 => R=1,2,3,4,5,6 #

And so #n(B)=12#, and s before #n(T)=36#, and so:

# P(B) = 12/36 = 1/3 #

And so we can now calculate:

# P( A | B) = (1/18) / (1/3) = 3/18 = 1/6 #

If green is #color(green)3# or #color(green)4#
and red is #color(red)1#, #color(red)2#, #color(red)3#, #color(red)4#, #color(red)5#, or #color(red)6#

#{: (,,,color(green)("green"),), (,ul("Sum of"),ul(" | "),ul(color(green)3),ul(color(green)4)), (color(red)("red"),color(red)1," | ",4,color(magenta)5), (,color(red)2," | ", color(magenta)5,6), (,color(red)3," | ", 6,7), (,color(red)4," | ", 7,8), (,color(red)5," | ", 8,9), (,color(red)6," | ", 9,10) :}#

As can be seen from the table:
#color(white)("XXX")#there are a total of #12# possible outcomes
and
#color(white)("XXX")# only #color(magenta)2# of those outcomes meet the requirement that the total be #5#

So the probability is #2/12 = 1/6#