Question

Two dice are thrown simultaneously. Find the probability that the score obtained is a perfect square or a prime number.

Answer 1

First of all, calculate the minimum and maximum score that can be obtained. The perfect square scores that are possible between the minimum and maximum scores are 4 and 9. Now, get all the possible outcomes so that the score is equal to 4 and 9. The prime numbers possible between the minimum and the maximum score are 2, 3, 5, 7, and 11. Now, get all the possible outcomes so that the scores are equal to 2, 3, 5, 7, and 11. The total number of outcomes possible for the first dice and the second dice is 6. Use the principle of multiplication and calculate the total number of possible outcomes when two dice are rolled. Solve it further and calculate the probability that the score obtained is a perfect square or a prime number by using the formula, Probability = $\dfrac{\text{No. of favorable outcomes}}{\text{Total no. of outcomes}}$.

Complete step-by-step solution
According to the question, it is given that two dice are thrown simultaneously and we are asked to find that the score obtained is a perfect square or a prime number.
First of all, let us find all the outcomes favorable for the perfect score.
Let us find the maximum and the minimum score possible when two dice are thrown simultaneously.
Here, the score is equal to the sum of the outcomes obtained from the first dice and the second dice.
For the minimum score, we can say that the outcomes for each dice should be minimum, which is equal to 1.
The minimum score = $1+1=2$ ……………………………….(1)
For the maximum score, we can say that the outcomes for each dice should be maximum, which is equal to 6.
The maximum score = $6+6=12$ ……………………………….(2)
So, our score must lie between the minimum and maximum scores.
We can see that the perfect square score between the maximum and the minimum score are 4, and 9.
For the score to be equal to 4, we have the following cases:
1 as an outcome for the first dice and 3 as an outcome for the second dice ………………………………………..(3)
2 as an outcome for the first dice and 2 as an outcome for the second dice ………………………………………..(4)
3 as an outcome for the first dice and 1 as an outcome for the second dice ………………………………………..(5)
From equation (3), equation (4), and equation (5), we have favorable outcomes for the first and the second dice to obtain the score 4.
So, the favorable outcomes for score equal to 4 are (1,3), (2,2), and (3,1) ……………………………………..(6)
Similarly, for the score to be equal to 9, we have the following cases:
3 as an outcome for the first dice and 6 as an outcome for the second dice ………………………………………..(7)
4 as an outcome for the first dice and 5 as an outcome for the second dice ………………………………………..(8)
5 as an outcome for the first dice and 4 as an outcome for the second dice ………………………………………..(9)
6 as an outcome for the first dice and 3 as an outcome for the second dice ………………………………………..(10)
From equation (7), equation (8), equation (9), and equation (10), we have favorable outcomes for the first and the second dice to obtain the score equal to 9.
So, the favorable outcomes for score equal to 9 are (3,6), (4,5), (5,4), and (6,3) ……………………………………..(11)
From equation (6) and equation (11), we have all possible cases to obtain a perfect square score.
The favorable outcomes for the perfect square score are (1,3), (2,2), (3,1), (3,6), (4,5), (5,4), and (6,3) ………………………………………..(12)
Now, let us find all the outcomes favorable for a prime number score.
From equation (1) and equation (2), we have the minimum and maximum score possible.
So, whenever two dice are rolled, our score would lie between the minimum and maximum score.
We can see that the prime number score between the minimum and the maximum score is 2, 3, 5, 7, and 11.
For the score to be equal to 2, we have the following case:
1 as an outcome for the first dice and again 1 as an outcome for the second dice ………………………………………(13)
For the score to be equal to 3, we have the following cases:
1 as an outcome for the first dice and 2 as an outcome for the second dice ……………………………………(14)
2 as an outcome for the first dice and 1 as an outcome for the second dice ……………………………………(15)
For the score to be equal to 5, we have the following cases:
1 as an outcome for the first dice and 4 as an outcome for the second dice ……………………………………(16)
2 as an outcome for the first dice and 3 as an outcome for the second dice ……………………………………(17)
3 as an outcome for the first dice and 2 as an outcome for the second dice ……………………………………(18)
4 as an outcome for the first dice and 1 as an outcome for the second dice ……………………………………(19)
For the score to be equal to 7, we have the following cases:
1 as an outcome for the first dice and 6 as an outcome for the second dice ……………………………………(20)
2 as an outcome for the first dice and 5 as an outcome for the second dice ……………………………………(22)
3 as an outcome for the first dice and 4 as an outcome for the second dice ……………………………………(23)
4 as an outcome for the first dice and 3 as an outcome for the second dice ……………………………………(24)
5 as an outcome for the first dice and 2 as an outcome for the second dice ……………………………………(25)
6 as an outcome for the first dice and 1 as an outcome for the second dice ……………………………………(26)
For the score to be equal to 11, we have the following cases:
5 as an outcome for the first dice and 6 as an outcome for the second dice ……………………………………(27)
6 as an outcome for the first dice and 5 as an outcome for the second dice ……………………………………(28)
From equation (13) to equation (28), we have all the possible aces favorable for the prime number score.
The favorable outcomes for prime number score are (1,1), (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (4,1), (4,3), (5,2), (5,6), (6,1) and (6,5) ………………………………………..(29)
Now, from equation (12) and equation (29), we have all the favorable outcomes for the perfect square and prime number score.
The total number of favorable outcomes for the perfect square or prime number score =
$7+15=22$ ………………………………………(30)
The total number of outcomes possible for the first dice = 6 ……………………………….(31)
The total number of outcomes possible for the second dice = 6 …………………………….(32)
Now, using the principle of multiplication, we get
The total number of outcomes possible for the first dice and second dice = $6\times 6=36$ ……………………………………(33)
We know the formula for the probability, Probability = $\dfrac{no.\,of\,favorable\,outcomes}{total\,no.\,of\,outcomes}$ …………………………………..(34)
Now, from equation (30), equation (33), and equation (34), we get
Probability = $\dfrac{22}{36}=\dfrac{11}{18}$ .
Hence, the probability that the score obtained is a perfect square or a prime number is $\dfrac{11}{18}$.

Note: In this question, one thing is hidden, that is here, the score is obtained by the summation of the outcomes obtained from the first dice and the second dice. One thing should be kept in mind, i.e, prime numbers are those which are divisible by 1 and itself.