$\frac{15}{36}$
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
- If one event is unaffected by the outcome of another event, the two events are said to be:
1. Dependent
2. Independent
3. Mutually exclusive
4. Mutually inclusive - Let $M$ be a randomly chosen non-empty subset of $S = \{1, 2, 3, \dots, 2026\}$.
Which of the following is the probability that the product of all the elements of $M$ is even? - A fair coin is tossed 6 times. The probability of getting at least 5 heads is _________________. (rounded off to two decimal places)
- Consider a random experiment of rolling a fair die. The probability of getting an odd number or a number less than 4 is ______ (Rounded off to two decimal places).
- In a null hypothesis significance test, the significance level $\alpha$ was set to 0.01. For which one or more of the following $p$ values would the null hypothesis be rejected?
