two fair dices are rolled simultaneously and it's outcome is noted.if the outcomes occurred on both the diced were summed up,then find the probability of getting a sum which is a prime number?
Answers
Answer:
The prime numbers between 2 to 12 are: 2,3,5,7,11.
The possibility for the sum 2 of the two thrown dices is 1/6 for “1” on the first dice multiply by 1/6 for “1” on the second dice - so 1/36 for the sum 2.
The possibility for sum 3 is only by the result of “1 2” - one dice “1” and second dice “2”, or vice versa (one dice “2” and second dice “1”). So it’s 2•(1/36) = 2/36 = 1/18 for the dices sum to 3.
The possibility of the sum 5 on the two dices is the possibility of rolling 1 4 or vice versa (4 1) or 2 3 or vice versa (3 2); So 4 separated combinations - each one of them with the chance of 1/36 - so 4•(1/36) = 4/36 =2/18 = 1/9
The possibility of sum 7 of the two dices is the possibility of rolling 1 6 or vice versa, 2 5 or vice versa, 3 4 or vice versa; So 6 separated combinations - each one of them with the chance of 1/36 - so 6•(1/36) = 6/36 = 3/18 = 1/6
The possibility of sum 11 is the possibility of sum 3 - which we have calculated earlier - due to obvious symmetry considerations (the possibility of rolling 5 6 or 6 5 is exactly the same as the possibility of rolling 1 2 or 2 1 of two fair dices). So it’s 2/36=1/18 as well.
Now we need to some all the odds in each main option because all of these outcomes are separated (parallel) possibility cases: so 1/36 + 2/36 + 4/36 + 6/36 + 2/36 = 15/36 = 5/12.
So the answer for your question is 5/12 which is 41.667% and not far from 1/2 - 50%.
If you used 2 pyramid fair dices - each one of them for 1 to 4 - the probability result for your question would be: first the only prime sums were 2,3,5 and 7
For sum 2: 1 1 - (1/4)•(1/4) = 1/16
For sum 3: 1 2 or 2 1 - 2•(1/16) = 2/16 = 1/8
For sum 5: 1 4 or 4 1 or 2 3 or 3 2 - 4•(1/16) = 4/16 = 1/4
For sum 7: the same prob. results such as sum - 3 from obvious symmetry considerations; So 2/16=1/8 as well.
Let’s sum them up all together: 1/16 + 2/16 + 4/16 + 2/16 = 9/16 - which is 56.25% - which is a bit more than 1/2 - 50%.
So in case of 1 to 4 pyramid fair dices - the probability of prime sum is a bit more than 1/2; and for 1 to 6 fair dices the probability is a little less than 1/2.