A Pair Of Dice Is Rolled And The Sum Of The Numbers Is Noted. What Is The Probability That The Sum Of 8 Does Not Occur?
When two dice are thrown what is the probability that the sum is 8.
The answer posted above is correct. However the same answer can be arrived at in another way also.
Tho get a total of 8 for 2 dies none of the dice must have a '1'.
The probability that the first dice will not have a '1', or that it will have 2, 3, 4, 5, or 6, are:
Probability of first dice not getting 1 = 5/6
This is because the probability of getting any number from 1 to 6 are equal.
For each occurrence of the number 2 to 6 for dice the second dice must get a number given by:
Required number if dice 2 = 8 - n
n = number of dice 1
As the probability of getting any one number is equal to 1/6, the probability that the second dice will have exactly the required number is:
Probability of second dice getting the number (8 - n) = 1/6
The probability of getting sum of 8:
Probability that sum is 8 =
(Probability of first dice not getting 1)*(Probability of second dice getting the number (8 - n))
= (5/6)*(1/6) = 5/36
Probability of sum of two dice as 8 = 5/36
We assume that each face of the dice number is numbered serially from 1 ,2,3,4,5,6.
When the two dice are thrown , one if the 36 ordered pairs can occur:
(1 ,1) , (1,2)......(1,6),
(2,1) , (2, 2) ....(2,6),
(3,1) , (3,2) ......(3,6)
(6,1) , (6,2) ........(6,6)
So a total of 8 can occur in follwing pairs:
(2 ,6) , (3,5) ,(4,4) , (5 , 3) ,(6, 2)
So there are 5 pairs of occurence of dice showing up faces when the serial number on the faces adds up 8, as against 36 possible pairs of numbers showing up.
So the probability of the event of showing the sum of the numbers on the face equal to 8 is 5/36.
We can get 8 when both the dice have 4, or when either has 6 and the other has 2 or when either has 5 and the other has 3. Also, here we use the property that the probability that two events take place together is the product of their individual probabilities.
Now the probability of any number when a die is thrown is 1/6. Let’s take the five cases we have here:
Both the dice have a 4: the probability that this happens is 1/6 * 1/6 = 1/36
The first die has a 5 and the second has a 3: the probability is 1/6 * 1/6 = 1/36
The first die has a 3 and the second has a 5: the probability is 1/6 * 1/6 = 1/36
The first die has a 6 and the second has a 2: the probability is 1/6 * 1/6 = 1/36
The first die has a 2 and the second has a 6: the probability is 1/6 * 1/6 = 1/36
Now we can get 8 if either of the five cases described above takes place. So the probability is the sum of the five individual probabilities which is 5*(1/36)= 5/36.
Therefore the probability that we get the sum as 8 when two dice are thrown is 5/36.