If two dice are rolled and their faces noted, find each of the following probabilities:
a) the sum of the two dice is 8
b) both dice show even numbers
c) the sum of the two dice is 8 OR both dice show even numbers
d) the sum of the numbers is 7, given that at least one die shows a 5
e) the sum of the numbers is 7, given that first die shows a 5
f) the sum is greater than 8 if the first die is a 6
Thanks
To solve these problems, we first need to determine the sample space, which is the set of all possible outcomes when rolling two dice. Each die has six possible outcomes (the numbers 1 to 6), so when rolling two dice, there are 6 * 6 = 36 possible outcomes.
Now let's find the probabilities for each problem:
a) The sum of the two dice is 8. To find this probability, we count the number of outcomes where the sum is 8. These outcomes are (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2), five outcomes in total. Therefore, the probability is 5/36.
b) Both dice show even numbers. To find this probability, we count the number of outcomes where both dice show even numbers. These outcomes are (2, 2), (2, 4), (2, 6), (4, 2), (4, 4), (4, 6), (6, 2), (6, 4), (6, 6), nine outcomes in total. Therefore, the probability is 9/36, which simplifies to 1/4.
c) The sum of the two dice is 8 OR both dice show even numbers. To find this probability, we need to find the outcomes that satisfy either condition. We've already counted the outcomes for both conditions in parts a) and b), so we just count the unique outcomes, which are (2, 6), (3, 5), (4, 4), (5, 3), (6, 2), (2, 2), (2, 4), (4, 6), (6, 4), and (6, 6), ten outcomes in total. Therefore, the probability is 10/36, which simplifies to 5/18.
d) The sum of the numbers is 7, given that at least one die shows a 5. To find this probability, we need to find the outcomes where the sum is 7 and at least one die shows a 5. The outcomes satisfying both conditions are (1, 6), (2, 5), (3, 4), (4, 3), and (5, 2), five outcomes in total. However, we need to consider only the outcomes where at least one die shows a 5, so we must exclude (1, 6) and (5, 2) since they both have a 5 and satisfy the sum condition. Therefore, the number of desired outcomes is three. By considering only the outcomes where at least one die shows a 5, we have reduced the sample space from 36 to 11 (subtracting the outcomes where no die shows a 5). Hence, the probability is 3/11.
e) The sum of the numbers is 7, given that the first die shows a 5. To find this probability, we need to find the outcomes where the sum is 7 and the first die shows a 5. The outcomes satisfying both conditions are (5, 2) and (5, 6), two outcomes in total. Since the first die always shows a 5, we have reduced the sample space to only 6 (the possible outcomes for the second die), so the probability is 2/6, which simplifies to 1/3.
f) The sum is greater than 8 if the first die is a 6. To find this probability, we need to find the outcomes where the sum is greater than 8 and the first die shows a 6. The outcomes satisfying both conditions are (6, 4), (6, 5), and (6, 6), three outcomes in total. Since the first die always shows a 6, the sample space is reduced to only 6 (the possible outcomes for the second die), so the probability is 3/6, which simplifies to 1/2.
I hope this helps! If you have any further questions, feel free to ask.
To find the probabilities, we need to determine the number of favorable outcomes and the total number of possible outcomes for each scenario.
a) The sum of the two dice is 8:
There are five ways to obtain a sum of 8: (2, 6), (3, 5), (4, 4), (5, 3), and (6, 2).
The total number of possible outcomes when rolling two dice is 36 (6 options for the first die and 6 options for the second die).
Therefore, the probability is 5/36.
b) Both dice show even numbers:
For each roll of a die, there are three possibilities for rolling an even number (2, 4, or 6). Since we have two dice, the total number of favorable outcomes is 3*3 = 9.
The total number of possible outcomes when rolling two dice is 36.
Therefore, the probability is 9/36, which simplifies to 1/4.
c) The sum of the two dice is 8 OR both dice show even numbers:
To find this probability, we need to add the probabilities from parts (a) and (b) but subtract the case where both dice show even numbers and the sum is 8 (which was counted twice):
P(sum of 8) = 5/36
P(both even numbers) = 1/4
P(both even numbers and sum of 8) = 1/36 (the only case is (4,4))
P(8 or both even) = P(sum of 8) + P(both even numbers) - P(both even numbers and sum of 8)
= 5/36 + 1/4 - 1/36
= 19/36
d) The sum of the numbers is 7, given that at least one die shows a 5:
In this case, there are three possible outcomes where the sum is 7: (1, 6), (2, 5), and (3, 4).
Out of these three cases, only (2, 5) meets the condition of having at least one die show a 5.
Therefore, the total number of favorable outcomes is 1.
The total number of possible outcomes when rolling two dice is 36.
Therefore, the probability is 1/36.
e) The sum of the numbers is 7, given that the first die shows a 5:
In this case, there is only one possible outcome where the sum is 7: (5, 2).
Therefore, the total number of favorable outcomes is 1.
The total number of possible outcomes when rolling two dice is 6 (since we are only concerned with the first die being a 5).
Therefore, the probability is 1/6.
f) The sum is greater than 8 if the first die is a 6:
There are two options for the second die when the sum is greater than 8: 3 and 4.
Therefore, the total number of favorable outcomes is 2.
The total number of possible outcomes when rolling two dice is 6 (since we are only concerned with the first die being a 6).
Therefore, the probability is 2/6, which simplifies to 1/3.
I will start you out.
a) There are 36 possibilities. 6,2; 2,6; 3,5; 5,3 are the only combinations that = 8.
b) P(even) = 3/6 = 1/2
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
c) Either-or probabilities are found by adding the individual probabilities.