Two fairly six~sided dice are rolled,find the probability that their total score is (a)2 (b)7 (c)a prime number (d)a perfect square. Solutions. (a)2/6=1/3 (b)7/6
There are 36 ways for the 2 dice to fall
(make yourself a 6by6 matrix)
a)
there is only one way to get a sum of 2 ---> 1,1
prob (a sum of 2) = 1/36
b)
a sum of 7:
1 6, 2 5, 3 4, 4 3, 5 2, 6 1, so 6 ways
prob(a sum of 7) = 6/36 = 1/6
c) a prime:
the primes are 2, 3, 5, 7, or 11
number of ways to get a 2: 1
number of ways to get a 3: 2
number of ways to get a 5: 4
etc
prob(a prime) = 15/36
another way is to look at your matrix from above, and count how many prime numbers you see
d) count the perfect squares in your matrix
perfect squares possible are 4 and 9
I count 7of them
prob(perfect square as the sum) = 7/36
To find the probability for each scenario, we need to determine the number of favorable outcomes (i.e., the number of ways to achieve the desired result) and divide it by the total number of possible outcomes.
(a) To find the probability of obtaining a total score of 2, we need to determine how many ways we can get a sum of 2 when rolling two six-sided dice. There is only one way to achieve this outcome, which is rolling a 1 on each die. The total number of possible outcomes when rolling two dice is 6*6 = 36 (since each die has 6 possible outcomes). Therefore, the probability is 1/36.
(b) To find the probability of obtaining a total score of 7, we need to determine how many ways we can get a sum of 7 when rolling two dice. The favorable outcomes are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). There are six favorable outcomes. Again, the total number of possible outcomes is 36. Therefore, the probability is 6/36, which simplifies to 1/6.
(c) To find the probability of obtaining a prime number as the total score, we need to identify the favorable outcomes. The possible sums when rolling two dice range from 2 to 12. The prime numbers in this range are 2, 3, 5, 7, 11. Among these prime numbers, only 2 and 3 can be achieved by rolling two dice. So, there are two favorable outcomes, and the probability is 2/36, which simplifies to 1/18.
(d) To find the probability of obtaining a perfect square as the total score, we need to identify the favorable outcomes. The possible sums when rolling two dice range from 2 to 12. The perfect squares in this range are 4 and 9. Both of these outcomes can be achieved by rolling two dice (e.g., (2, 2), (3, 1), (1, 3), (2, 2)). Therefore, there are four favorable outcomes, and the probability is 4/36, which simplifies to 1/9.