Two six-sided dice are rolled at the same time and the numbers showing are observed. Find the following.
a. P(sum=2)
b. P(sum=3)
c. P(sum is an even number)
d. P (the two dice show different numbers)
there are 36 possible outcomes ... 6 on each die
list them and count the probabilities
To find the probabilities for each scenario, we need to determine the possible outcomes and calculate the number of favorable outcomes for each event.
a. P(sum=2):
There is only one way to roll a sum of 2, which is when both dice show a 1. The favorable outcome is 1, and there are a total of 36 possible outcomes (6 possible outcomes for each die, as there are 6 sides on a die). Therefore, the probability is:
P(sum=2) = favorable outcomes / possible outcomes = 1 / 36
b. P(sum=3):
There are two ways to roll a sum of 3: (1, 2) or (2, 1). The favorable outcomes are 2, and the total possible outcomes remain 36. Therefore, the probability is:
P(sum=3) = favorable outcomes / possible outcomes = 2 / 36 = 1 / 18
c. P(sum is an even number):
To calculate the probability of rolling a sum that is an even number, we need to determine the favorable outcomes. There are a total of 36 possible outcomes, and out of these, 18 outcomes have an even sum: (1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (2, 6), (3, 1), (3, 3), (3, 5), (4, 2), (4, 4), (4, 6), (5, 1), (5, 3), (5, 5), (6, 2), (6, 4), and (6, 6). Therefore, the probability is:
P(sum is an even number) = favorable outcomes / possible outcomes = 18 / 36 = 1 / 2
d. P(the two dice show different numbers):
To find the probability that the two dice show different numbers, we need to calculate the favorable outcomes. There are a total of 36 possible outcomes, and out of these, 30 outcomes have the two dice showing different numbers: (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5). Therefore, the probability is:
P(the two dice show different numbers) = favorable outcomes / possible outcomes = 30 / 36 = 5 / 6
To find the probabilities of different outcomes when rolling two six-sided dice, we first need to determine the total number of possible outcomes.
Each die has 6 possible outcomes, so the total number of outcomes when rolling two dice simultaneously is 6 x 6 = 36.
Now, let's find the probabilities for each part of the question:
a. P(sum=2)
To find the probability of getting a sum of 2, we need to determine the number of outcomes that satisfy this condition. There is only 1 outcome where the sum is 2: when both dice show a 1.
So, P(sum=2) = 1/36.
b. P(sum=3)
To find the probability of getting a sum of 3, we need to determine the number of outcomes that satisfy this condition. There are 2 outcomes where the sum is 3: (1,2) and (2,1).
So, P(sum=3) = 2/36 = 1/18.
c. P(sum is an even number)
To find the probability of getting a sum that is an even number, we need to determine the number of outcomes that satisfy this condition. There are 18 outcomes where the sum is even: (1,1), (1,3), (1,5), (2,2), (2,4), (2,6), (3,1), (3,3), (3,5), (4,2), (4,4), (4,6), (5,1), (5,3), (5,5), (6,2), (6,4), and (6,6).
So, P(sum is an even number) = 18/36 = 1/2.
d. P(the two dice show different numbers)
To find the probability of getting two dice that show different numbers, we need to determine the number of outcomes that satisfy this condition. There are 30 outcomes where the two dice show different numbers: (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5).
So, P(the two dice show different numbers) = 30/36 = 5/6.