A pair of dice is rolled, one black and one white. Find the probability of each of the following events.
1. The total is between 3 and 7.
My answer: 1/3
2. The total is 13.
My answer: 0
3. The numbers are 2 and 5.
?????????
4. The black die has 2 and the white die has 5.
??????
5. The black die has 2 or the white die has 5.
??????
Make block matrix 6 by 6 for total of 36 possible outcomes. Then count how many satisfy your criteria.
1. I get 20/36 = 5/9
2. agree
3. 2/36 = 1/18
4. 1/6 * 1/6 = 1/36
5. 1/6 + 1/6 = 1/3
Well, let's roll these dice and see what happens!
1. The total is between 3 and 7. So we need the sum to be either 3, 4, 5, 6, or 7. Let's list the possibilities:
- 1 + 2 = 3
- 2 + 1 = 3
- 1 + 3 = 4
- 3 + 1 = 4
- 1 + 4 = 5
- 4 + 1 = 5
- 2 + 3 = 5
- 3 + 2 = 5
- 1 + 5 = 6
- 5 + 1 = 6
- 2 + 4 = 6
- 4 + 2 = 6
- 3 + 3 = 6
- 1 + 6 = 7
- 6 + 1 = 7
- 2 + 5 = 7
- 5 + 2 = 7
- 3 + 4 = 7
- 4 + 3 = 7
- 6 + 2 = 8
- 2 + 6 = 8
- 3 + 5 = 8
- 5 + 3 = 8
- 4 + 4 = 8
- 6 + 3 = 9
- 3 + 6 = 9
- 4 + 5 = 9
- 5 + 4 = 9
- 6 + 4 = 10
- 4 + 6 = 10
- 6 + 5 = 11
- 5 + 6 = 11
- 6 + 6 = 12
Out of these 36 possible outcomes, there are 25 where the sum is between 3 and 7. So the probability is 25/36.
2. The total is 13. Hmm, we can't get a sum of 13 with just two dice. Unless... one of the dice is a cheeseburger and the other is a donut, and they magically combine to form a sum of 13! But in this case, since we only have regular dice, the probability is 0.
3. The numbers are 2 and 5. Well, we just need the black die to show a 2 and the white die to show a 5. This can happen in only 1 out of the 36 possible outcomes. So the probability is 1/36. It's a pretty rare event, I must say!
4. The black die has 2 and the white die has 5. We're looking for just one specific outcome out of the 36 possibilities. So the probability is 1/36. It's like trying to find your car keys in a haystack!
5. The black die has 2 or the white die has 5. There are a few ways this could happen: the black die shows 2 and the white die could be anything, or the white die shows 5 and the black die could be anything. Let's list the possibilities:
- 2 + 1 = 3
- 2 + 2 = 4
- 2 + 3 = 5
- 2 + 4 = 6
- 2 + 5 = 7
- 2 + 6 = 8
- 1 + 5 = 6
- 3 + 5 = 8
- 4 + 5 = 9
- 5 + 5 = 10
- 6 + 5 = 11
Out of these 11 possibilities, 8 of them satisfy the condition. So the probability is 8/36, which simplifies to 2/9. Quite an unusual combination, if you ask me!
I hope that clarified things for you. If you have any more questions, feel free to ask!
To answer questions 3, 4, and 5, we will need to first calculate the total number of possible outcomes when rolling the black and white dice.
Question 3: The numbers are 2 and 5.
To calculate the probability of this specific outcome, we need to determine how many ways the black die can show a 2 and the white die can show a 5. Since there's only one die of each color, there is only one way for this specific outcome to occur. Therefore, the probability is 1/36.
Question 4: The black die has 2 and the white die has 5.
Similar to the previous question, there is only one way for the black die to show a 2 and the white die to show a 5. Therefore, the probability is 1/36.
Question 5: The black die has 2 or the white die has 5.
To calculate the probability of this event, we need to determine the number of ways the black die can show a 2, the number of ways the white die can show a 5, and subtract the overlap (since both dice cannot show their respective numbers simultaneously).
The black die can show a 2 in one way, and the white die can show a 5 in one way. Since these two events do not overlap, the probability of either the black die showing 2 or the white die showing 5 is 2/36, which simplifies to 1/18.
So, the answers are:
3. The probability that the numbers rolled are 2 and 5 is 1/36.
4. The probability that the black die shows 2 and the white die shows 5 is 1/36.
5. The probability that either the black die shows 2 or the white die shows 5 is 1/18.
To find the probability of each of the events, we need to determine the number of favorable outcomes and the total number of possible outcomes.
Let's go through each event step-by-step:
1. The total is between 3 and 7.
To find the probability of this event, we need to count the number of ways we can roll a total between 3 and 7, and then divide it by the total number of possible outcomes.
To roll a total between 3 and 7, we can have the following combinations:
- (1, 2)
- (2, 1)
- (1, 3)
- (3, 1)
- (1, 4)
- (4, 1)
- (2, 2)
- (1, 5)
- (5, 1)
- (2, 3)
- (3, 2)
The total favorable outcomes are 11. The total number of possible outcomes when rolling two dice is 36 (6 for the first die and 6 for the second die, giving a total of 6 * 6 = 36).
Therefore, the probability of rolling a total between 3 and 7 is 11/36.
2. The total is 13.
To find the probability of rolling a total of 13, we again need to count the number of ways we can roll this total and divide it by the total number of possible outcomes.
However, when rolling two dice, we can never get a total of 13 since the maximum sum we can achieve is 12 (6 from each die, which is the maximum value on a standard die).
Therefore, the probability of rolling a total of 13 is 0.
3. The numbers are 2 and 5.
To find the probability of rolling a 2 and a 5 on the two dice, we need to determine how many ways this combination can occur and divide it by the total number of possible outcomes.
In this case, there is only one favorable outcome - rolling a 2 on the black die and a 5 on the white die.
The total number of possible outcomes remains the same at 36.
Therefore, the probability of rolling a 2 and a 5 is 1/36.
4. The black die has 2 and the white die has 5.
To find the probability of the black die showing 2 and the white die showing 5, we need to determine how many ways this combination can occur and divide it by the total number of possible outcomes.
Similarly to the previous event, there is only one favorable outcome - the black die showing 2 and the white die showing 5.
Therefore, the probability of the black die showing 2 and the white die showing 5 is 1/36.
5. The black die has 2 or the white die has 5.
To find the probability of either the black die showing 2 or the white die showing 5, we can calculate the probability of each event individually and then sum them up.
- Probability of the black die showing 2:
The favorable outcome is 1 (the black die shows 2) and the total number of possible outcomes is 6.
Therefore, the probability of the black die showing 2 is 1/6.
- Probability of the white die showing 5:
The favorable outcome is 1 (the white die shows 5) and the total number of possible outcomes is again 6.
Therefore, the probability of the white die showing 5 is 1/6.
To find the probability of either of these events occurring, we sum up their individual probabilities: 1/6 + 1/6 = 2/6 = 1/3.
Therefore, the probability of either the black die showing 2 or the white die showing 5 is 1/3.