$\frac{15}{36}$
Understanding the Dice Probability Problem
The question asks for the probability of obtaining a prime number sum when rolling an unbiased six-sided dice (faces 1 to 6) twice in succession.
Determining Total Possible Outcomes
An unbiased six-sided dice has 6 possible outcomes for each roll (1, 2, 3, 4, 5, 6).
When rolling the dice twice, the total number of unique outcomes is calculated by multiplying the number of outcomes for each roll:
Total Possible Outcomes = $6 \times 6 = 36$
Identifying Prime Numbers within the Sum Range
The minimum sum achievable is $1 + 1 = 2$.
The maximum sum achievable is $6 + 6 = 12$.
The prime numbers between 2 and 12 (inclusive) are: 2, 3, 5, 7, and 11.
Counting Favorable Outcomes for a Prime Sum
We need to find all pairs of outcomes from the two rolls whose sum is one of these prime numbers.
| Prime Sum | Favorable Pairs (Roll 1, Roll 2) | Count |
| 2 | (1, 1) | 1 |
| 3 | (1, 2), (2, 1) | 2 |
| 5 | (1, 4), (4, 1), (2, 3), (3, 2) | 4 |
| 7 | (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3) | 6 |
| 11 | (5, 6), (6, 5) | 2 |
Summing the counts for each prime sum gives the total number of favorable outcomes:
Total Favorable Outcomes = $1 + 2 + 4 + 6 + 2 = 15$
Calculating the Final Probability
The probability is the ratio of favorable outcomes to the total possible outcomes.
Probability (Sum is Prime) = $\frac{\text{Total Favorable Outcomes}}{\text{Total Possible Outcomes}}$
Probability = $\frac{15}{36}$
Important Questions from Probability
Three dice are thrown. What is the probability of getting a sum which is a perfect square?
Two distinct natural numbers from 1 to 9 are picked at random. What is the probability that their product has 1 in its unit place?
Two dice are thrown. What is the probability that difference of numbers on them is 2 or 3 ?
Suppose that there is a chance for a newly constructed building to collapse, whether the design is faulty or not. The chance that the design is faulty is 10%. The chance that the building collapses is 95% if the design is faulty, otherwise it is 45%. If it is seen that the building has collapsed, then what is the probability that it is due to faulty design?
What is the probability that all three boys sit together?
