If you roll a pair of dice, what is the probability of rolling either a single 5 or a sum that is an even number?

Question

Nora Amin · Accepted Answer

#2/3# Note that total number of possible cases are #6^2= 36# Getting a single #5# ( Say event #A# ) means situation like as #(1,5),(2,5),(3,5),(4,5),(6,5),(5,1),(5,2),(5,3),(5,4),(5,6)# i.e. #10# cases and we have #18# cases when sum is an even number ( Say event #B# ) . But these two events are not mutually exclusive. Here #(1,5),(3,5),(5,1),(5,3)# i.e. #4# cases where we get a single #5# as well as sum is a even number ( Say event #A nn B# ) . So we have number of favorable cases to our event #= n(A)+n(B)-n(A nn B) = 10+18-4=24# So required probability #= 24/36=2/3#

Jameson Adams · Answer

undefined To find the probability of rolling either a single 5 or a sum that is an even number with a pair of dice, we can break it down into separate events.

1. Probability of rolling a single 5:
   There are 11 possible outcomes when rolling two dice and getting a single 5: (1,5), (2,5), (3,5), (4,5), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), and (6,5). Out of these, there are 10 outcomes that result in rolling a single 5. Therefore, the probability of rolling a single 5 is 10/36, or approximately 0.278.

2. Probability of rolling a sum that is an even number:
   To find the probability of rolling a sum that is an even number, we need to consider the combinations of dice rolls that result in even sums. An even sum can result from the following combinations: (1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3), (3,5), (4,2), (4,4), (4,6), (5,1), (5,3), (5,5), (6,2), (6,4), and (6,6). There are 18 possible outcomes that result in an even sum.
   
Therefore, the total probability of rolling either a single 5 or a sum that is an even number is the sum of the individual probabilities:

10/36 (rolling a single 5) + 18/36 (rolling an even sum) = 28/36, or approximately 0.778.