Two six-sided dice are rolled at the same time and the numbers showing are observed. Find the following.

a. P(sum=6)
(1,6), (6,1)
1/36+1/36=2/36


b. P(sum=3) =1/36



c. P(sum is an odd number)


30/36
d. P(the two dice show same numbers) ?
Can you check my answers and show me the correct steps for a-d.

Sure, I'll be happy to help! Let's go through each question step by step:

a. P(sum=6):
To find the probability of getting a sum of 6, we need to count all the possible outcomes that sum up to 6 and divide it by the total number of possible outcomes. In this case, the possible outcomes are (1,6) and (6,1). So, the probability is 2/36, which can be simplified to 1/18.

Your answer is correct!

b. P(sum=3):
To find the probability of getting a sum of 3, we need to count all the possible outcomes that sum up to 3 and divide it by the total number of possible outcomes. In this case, the only outcome is (1,2) or (2,1). So, the probability is 1/36.

Your answer is correct!

c. P(sum is an odd number):
To find the probability of getting a sum that is an odd number, we need to count all the possible outcomes that have an odd sum and divide it by the total number of possible outcomes. Out of the 36 possible outcomes, half of them have an odd sum (i.e., 18 outcomes). So, the probability is 18/36, which simplifies to 1/2.

Your answer is incorrect. It should be 1/2, not 30/36.

d. P(the two dice show the same numbers):
To find the probability of getting the same numbers on both dice, we need to count all the possible outcomes where both dice land on the same number and divide it by the total number of possible outcomes. There are 6 possible outcomes where both dice show the same number, which are (1,1), (2,2), (3,3), (4,4), (5,5), and (6,6). So, the probability is 6/36, which simplifies to 1/6.

Your answer is incorrect. It should be 1/6, not 1/36.

I hope this helps clarify things for you! Let me know if you have any more questions.

a. To find the probability of getting a sum of 6 when rolling two six-sided dice, you can list all the possible combinations and count how many of them have a sum of 6.

The possible combinations that give a sum of 6 are: (1,5), (2,4), (3,3), (4,2), and (5,1). So there are 5 possible combinations.

Since there are a total of 36 possible outcomes when rolling two dice (each die has 6 possible outcomes, so 6 x 6 = 36), the probability of getting a sum of 6 is:

P(sum=6) = 5/36.

b. Similarly, to find the probability of getting a sum of 3, you can list all the possible combinations and count how many of them have a sum of 3.

The only combination that gives a sum of 3 is (1,2). So there is only 1 possible combination.

Since there are a total of 36 possible outcomes when rolling two dice, the probability of getting a sum of 3 is:

P(sum=3) = 1/36.

c. To find the probability of getting a sum that is an odd number, you need to count how many possible combinations have a sum that is odd.

Out of the 36 possible combinations, the following combinations have an odd sum: (1,1), (1,3), (1,5), (2,1), (2,3), (2,5), (3,1), (3,3), (3,5), (4,1), (4,3), (4,5), (5,1), (5,3), (5,5), (6,1), (6,3), (6,5). So there are 18 possible combinations.

Therefore, the probability of getting a sum that is an odd number is:

P(sum is an odd number) = 18/36 = 1/2.

d. To find the probability of the two dice showing the same numbers, you need to count how many possible combinations have both dice showing the same numbers.

Out of the 36 possible combinations, the following combinations have the same numbers: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6). So there are 6 possible combinations.

Therefore, the probability of the two dice showing the same numbers is:

P(the two dice show same numbers) = 6/36 = 1/6.

Let's go through each question step by step and determine the correct answers.

a. P(sum=6)
To find the probability of getting a sum of 6 when rolling two six-sided dice, we need to count the number of favorable outcomes and divide it by the total number of possible outcomes.

Favorable outcomes: (1,6) and (6,1)
Total outcomes: Since each die has 6 possible outcomes, the total number of outcomes is 6 x 6 = 36.

So, the probability is: 2/36 = 1/18.

b. P(sum=3)
In order to obtain a sum of 3 when rolling two dice, only one outcome gives this result: (1,2) or (2,1). Therefore, the number of favorable outcomes is 1, and the total number of outcomes remains 36.

So, the probability is: 1/36.

c. P(sum is an odd number)
We can approach this problem by counting the number of favorable outcomes and dividing it by the total number of outcomes.

Favorable outcomes: (1,2), (1,4), (1,6), (2,1), (2,3), (2,5), (3,2), (3,4), (3,6), (4,1), (4,3), (4,5), (5,2), (5,4), (5,6), (6,1), (6,3), (6,5)
Total outcomes: 36

So, the probability is: 18/36 = 1/2.

d. P(the two dice show the same numbers)
To find the probability of both dice showing the same number, we count the number of favorable outcomes and divide that by the total number of outcomes.

Favorable outcomes: (1,1), (2,2), (3,3), (4,4), (5,5), (6,6)
Total outcomes: 36

So, the probability is: 6/36 = 1/6.

Double-checking your answers:
a. P(sum=6) - You correctly calculated the probability as 1/18.
b. P(sum=3) - You correctly calculated the probability as 1/36.
c. P(sum is an odd number) - You incorrectly calculated it as 30/36. The correct probability is 1/2 as explained above.
d. P(the two dice show same numbers) - You correctly calculated the probability as 1/6.

So, your answers for questions a, b, and d are correct, but your answer for question c is incorrect.

Out of 36 possibilities:

a. If the sum = 6, the two dice have to add up to 6.
(5,1)(1,5)(4,2) (2,4)(3,3)

b. (1,2)(2,1)

c. To sum an odd number, one dice needs to be odd, while the other is even. (1,2)(1,4)(1,6)(6,1)(4,1)(2,1)(3,2)....

d. (1,1)(2,2)(3,3)(4,4)(5,5)(6,6)