If two unbiased six faced dice are thrown, the probability that the sum of the numbers on both the faces turned up, is a prime number greater than 5 is:
Dice Roll Probability Explained
When two unbiased six-faced dice are thrown, we need to determine the probability that the sum of the numbers on both faces is a prime number greater than 5. Let's break this down step-by-step to understand the calculation.
Total Outcomes for Two Dice
First, we identify the total possible outcomes when two six-faced dice are thrown. Each die has 6 faces (numbered 1 to 6). Since the dice throws are independent, the total number of possible outcomes in the sample space is the product of the outcomes for each die.
- Number of outcomes for the first die = 6
- Number of outcomes for the second die = 6
So, the total number of possible outcomes is \(6 \times 6 = 36\). We can represent these outcomes as ordered pairs \((d_1, d_2)\) where \(d_1\) is the result of the first die and \(d_2\) is the result of the second die.
| Die 1 \(\downarrow\) / Die 2 \(\rightarrow\) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
Identifying Prime Sums
Next, we need to find the sums of the numbers on the faces that are prime numbers greater than 5. The minimum possible sum is \(1+1=2\), and the maximum possible sum is \(6+6=12\).
Let's list all prime numbers between 2 and 12:
- 2
- 3
- 5
- 7
- 11
From this list, we need to identify the prime numbers that are greater than 5. These are:
- 7
- 11
Outcomes for Sum Equal to 7
Now, let's list all the pairs of numbers from the two dice that sum up to 7:
- (1, 6)
- (2, 5)
- (3, 4)
- (4, 3)
- (5, 2)
- (6, 1)
There are 6 favorable outcomes where the sum is 7.
Outcomes for Sum Equal to 11
Next, let's list all the pairs of numbers from the two dice that sum up to 11:
- (5, 6)
- (6, 5)
There are 2 favorable outcomes where the sum is 11.
Combined Favorable Outcomes
The total number of favorable outcomes (where the sum is a prime number greater than 5) is the sum of outcomes for 7 and 11.
- Favorable outcomes for sum 7 = 6
- Favorable outcomes for sum 11 = 2
Total favorable outcomes = \(6 + 2 = 8\).
Final Probability Calculation
The probability of an event is calculated using the formula:
$$P(\text{Event}) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}}$$
Using the values we found:
- Number of Favorable Outcomes = 8
- Total Number of Outcomes = 36
So, the probability is:
$$P(\text{sum is a prime number greater than 5}) = \frac{8}{36}$$
To simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 4:
$$\frac{8 \div 4}{36 \div 4} = \frac{2}{9}$$
Therefore, the probability that the sum of the numbers on both the faces turned up is a prime number greater than 5 is \(\frac{2}{9}\).
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?
