Two dice are thrown simultaneously. What is the probability that the sum of the numbers appearing on them is a prime number?
Understanding the Problem: Probability with Two Dice
The question asks for the probability that the sum of the numbers shown on the faces of two dice, when thrown simultaneously, is a prime number. To solve this, we first need to determine the total possible outcomes when throwing two dice and then identify the outcomes where the sum is a prime number.
Calculating Total Outcomes
When a single die is thrown, there are 6 possible outcomes (1, 2, 3, 4, 5, 6). When two dice are thrown simultaneously, the outcome is an ordered pair representing the results on each die. The total number of possible outcomes is the product of the outcomes for each die.
Total number of outcomes = Outcomes on Die 1 × Outcomes on Die 2
Total number of outcomes = $6 \times 6 = 36$
These 36 outcomes form the sample space for this experiment. We can list them as pairs (Result on Die 1, Result on Die 2).
Identifying Favourable Outcomes: Sum is a Prime Number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The possible sums when rolling two dice range from $1+1=2$ to $6+6=12$. We need to find the prime numbers within this range.
The prime numbers between 2 and 12 (inclusive) are 2, 3, 5, 7, and 11.
Now, let's list the pairs of outcomes from the 36 total possibilities that result in these prime sums:
- Sum = 2: (1, 1) - 1 outcome
- Sum = 3: (1, 2), (2, 1) - 2 outcomes
- Sum = 5: (1, 4), (4, 1), (2, 3), (3, 2) - 4 outcomes
- Sum = 7: (1, 6), (6, 1), (2, 5), (5, 2), (3, 4), (4, 3) - 6 outcomes
- Sum = 11: (5, 6), (6, 5) - 2 outcomes
The total number of favourable outcomes (where the sum is a prime number) is the sum of the outcomes for each prime sum:
Total favourable outcomes = $1 + 2 + 4 + 6 + 2 = 15$
Calculating the Probability
The probability of an event is calculated as the ratio of the number of favourable outcomes to the total number of possible outcomes.
Probability (Sum is a prime number) = $\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$
Probability = $\frac{15}{36}$
This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
Probability = $\frac{15 \div 3}{36 \div 3} = \frac{5}{12}$
Thus, the probability that the sum of the numbers appearing on two simultaneously thrown dice is a prime number is $\frac{5}{12}$.
Revision Table: Probability Concepts
| Concept | Description | Formula/Example |
|---|---|---|
| Probability | A measure of the likelihood of an event occurring. | $P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}$ |
| Sample Space | The set of all possible outcomes of a random experiment. | For rolling one die, S = {1, 2, 3, 4, 5, 6} |
| Event | A subset of the sample space. | Rolling an even number on a die: {2, 4, 6} |
| Prime Number | A natural number greater than 1 with only two divisors: 1 and itself. | 2, 3, 5, 7, 11, 13, etc. |
Additional Information: Two Dice Outcomes
When rolling two dice, there are 36 possible outcomes. We can list them in a table to visualize the sample space and their sums:
| Die 1 \ Die 2 | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | (1,1) Sum=2 | (1,2) Sum=3 | (1,3) Sum=4 | (1,4) Sum=5 | (1,5) Sum=6 | (1,6) Sum=7 |
| 2 | (2,1) Sum=3 | (2,2) Sum=4 | (2,3) Sum=5 | (2,4) Sum=6 | (2,5) Sum=7 | (2,6) Sum=8 |
| 3 | (3,1) Sum=4 | (3,2) Sum=5 | (3,3) Sum=6 | (3,4) Sum=7 | (3,5) Sum=8 | (3,6) Sum=9 |
| 4 | (4,1) Sum=5 | (4,2) Sum=6 | (4,3) Sum=7 | (4,4) Sum=8 | (4,5) Sum=9 | (4,6) Sum=10 |
| 5 | (5,1) Sum=6 | (5,2) Sum=7 | (5,3) Sum=8 | (5,4) Sum=9 | (5,5) Sum=10 | (5,6) Sum=11 |
| 6 | (6,1) Sum=7 | (6,2) Sum=8 | (6,3) Sum=9 | (6,4) Sum=10 | (6,5) Sum=11 | (6,6) Sum=12 |
Looking at the sums in the table, we can count the occurrences of prime sums (2, 3, 5, 7, 11):
- Sum 2: 1 time
- Sum 3: 2 times
- Sum 5: 4 times
- Sum 7: 6 times
- Sum 11: 2 times
Total prime sums = $1 + 2 + 4 + 6 + 2 = 15$. This confirms our count of favourable outcomes.
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?
