probability of rolling two dice and rolling a sum of 8 or a multiple of 2 on one of the dice
To find the probability of these events occurring, we need to determine the number of favorable outcomes (i.e., rolling an 8 or a multiple of 2) and divide it by the total number of possible outcomes.
The total number of possible outcomes when rolling two dice is 36 (6 possible outcomes for the first die multiplied by 6 possible outcomes for the second die).
Now let's determine the number of favorable outcomes for each event:
1. Rolling a sum of 8: There are 5 favorable outcomes for this event, namely (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2). This is because the sum of the two dice must be 8, and any combination that satisfies this condition is a favorable outcome.
2. Rolling a multiple of 2 on one of the dice: There are 18 favorable outcomes for this event. This is because out of the 36 possible outcomes, exactly half (18) have at least one die that is a multiple of 2. Note that this includes cases where both dice are multiples of 2, which we have already counted in the first event.
Now, let's calculate the probability of each event:
1. Probability of rolling a sum of 8: 5 favorable outcomes divided by 36 total outcomes equals 5/36.
2. Probability of rolling a multiple of 2 on one of the dice: 18 favorable outcomes divided by 36 total outcomes equals 1/2.
Finally, since we are interested in the probability of either event occurring, we add their probabilities together:
Probability of rolling a sum of 8 or a multiple of 2 on one of the dice = 5/36 + 1/2 = (5 + 18) / 36 = 23/36.
Therefore, the probability of rolling a sum of 8 or a multiple of 2 on one of the dice is 23/36.
how are there 36 possible outcomes
Apologies for any confusion in my response. You are correct, there are not 36 possible outcomes when rolling two dice. The total number of possible outcomes is actually 36, as there are 6 possible outcomes for the first die and 6 possible outcomes for the second die (6 x 6 = 36).
To find the number of favorable outcomes, we need to determine the specific combinations that satisfy each event:
1. Rolling a sum of 8: There are 5 favorable outcomes for this event: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2).
2. Rolling a multiple of 2 on one of the dice: There are 21 favorable outcomes for this event. This includes the following combinations: (2, any number), (4, any number), (6, any number), (any number, 2), (any number, 4), and (any number, 6). There are a total of 21 combinations that satisfy this condition.
Now, let's calculate the probability of each event:
1. Probability of rolling a sum of 8: 5 favorable outcomes divided by 36 total outcomes equals 5/36.
2. Probability of rolling a multiple of 2 on one of the dice: 21 favorable outcomes divided by 36 total outcomes equals 21/36.
Lastly, to find the probability of either event occurring, we add their probabilities together:
Probability of rolling a sum of 8 or a multiple of 2 on one of the dice = 5/36 + 21/36 = 26/36.
Therefore, the correct probability of rolling a sum of 8 or a multiple of 2 on one of the dice is 26/36.