$\frac{15}{36}$
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
Two dice are thrown simultaneously. What is the probability of getting the same number on both the dice?
The probability of being 53 Sundays in year 2020 is-
Three dice are thrown randomly. The probability of coming 3 in at least one die is
The probability of having 53 Tuesdays in an ordinary year is:
When two dice are tossed simultaneously, the probability that the sum of the numbers appearing on both the dice is 8 will be
