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.
??????
#1: I get 20/36 = 5/9
B:1 & W:2,3,4,5,6
B:2 & W:1,2,3,4,5
B:3 & W:1,2,3,4
B:4 & W:1,2,3
B:5 & W:1,2
B:6 & W:1
#2 ok
#3: 2/36 = 1/18
B:2 & W:5
B:5 & W:2
#4: 1/6 * 1/6 = 1/36
#5: 1/6 + 1/6 = 1/3
You can list all 12 events if you want
3. The probability of rolling a 2 on the black die and a 5 on the white die is 1/36 since there is only one way to achieve this outcome (the black die rolling a 2 and the white die rolling a 5) out of the 36 possible outcomes when rolling two dice.
4. The probability of the black die rolling a 2 and the white die rolling a 5 is 1/36 since there is only one way for this exact combination to occur.
5. The probability of the black die rolling a 2 or the white die rolling a 5 is 11/36. There are 11 possibilities out of the 36 possible outcomes where either the black die rolls a 2 or the white die rolls a 5, or both.
To find the probability of each of the given events, let's break down each question step-by-step:
1. The total is between 3 and 7.
To calculate this probability, we need to count the total number of outcomes where the sum of the two dice is between 3 and 7.
There are 6 possible outcomes for the black die (1, 2, 3, 4, 5, or 6) and 6 possible outcomes for the white die (1, 2, 3, 4, 5, or 6).
Out of these possible combinations, there are 10 combinations that have a sum between 3 and 7:
(1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (4, 1), (5, 1).
Therefore, the probability of the total being between 3 and 7 is 10/36, which simplifies to 5/18.
2. The total is 13.
To find the probability of the total being 13, we need to count the number of outcomes where the sum of the two dice is equal to 13.
There is only one combination (6, 7) that results in a sum of 13.
Since there is only one favorable outcome out of a total of 36 possible outcomes, the probability of the total being 13 is 1/36.
3. The numbers are 2 and 5.
To find the probability of getting a 2 on the black die and a 5 on the white die, we need to find the probability of each die landing on the desired number and multiply them together.
The probability of getting a 2 on the black die is 1/6.
The probability of getting a 5 on the white die is also 1/6.
Multiplying these probabilities together, we get (1/6) * (1/6) = 1/36.
Therefore, the probability of getting a 2 on the black die and a 5 on the white die 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 multiply the probabilities of each die landing on the desired number.
The probability of the black die landing on 2 is 1/6.
The probability of the white die landing on 5 is also 1/6.
Multiplying these probabilities together, we get (1/6) * (1/6) = 1/36.
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 need to add the probabilities of each event occurring.
The probability of the black die showing 2 is 1/6.
The probability of the white die showing 5 is 1/6.
Adding these probabilities together, we get 1/6 + 1/6 = 2/6, which simplifies to 1/3.
Therefore, the probability of either the black die showing 2 or the white die showing 5 is 1/3.
I hope this helps! Let me know if you have any further questions.
To find the probability of each event, we need to determine the total number of possible outcomes and the number of favorable outcomes for each event. Let's consider each event one by one:
1. The total is between 3 and 7.
To find the total number of possible outcomes, we need to consider the number of distinct outcomes when rolling two dice. Each die can take on 6 different values (numbers 1 to 6), so the total number of outcomes is 6 * 6 = 36.
Now, let's determine the number of favorable outcomes. The ways to get a total between 3 and 7 are:
- 1 + 2
- 2 + 1
- 1 + 3
- 3 + 1
- 1 + 4
- 4 + 1
- 1 + 5
- 5 + 1
- 2 + 3
- 3 + 2
- 2 + 4
- 4 + 2
- 2 + 5
- 5 + 2
There are 14 favorable outcomes. Therefore, the probability of the total being between 3 and 7 is 14/36, which simplifies to 7/18.
2. The total is 13.
The sum of the dice can never be 13 since the maximum value on a die is 6. Therefore, the number of favorable outcomes is 0. The probability of the total being 13 is 0/36, which simplifies to 0.
3. The numbers are 2 and 5.
To find the probability of both dice showing a 2 and a 5, we need to consider the same process as before. The total number of outcomes is 36, and there is only one favorable outcome: the black die showing 2 and the white die showing 5. Therefore, the probability is 1/36.
4. The black die has 2 and the white die has 5.
To determine the probability of this event, we consider the total number of outcomes (36) and the number of favorable outcomes. There is only one favorable outcome: the black die showing 2 and the white die showing 5. Therefore, the probability 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 need to find the number of favorable outcomes.
The favorable outcomes are:
- Black die shows 2 and white die shows any number (5 options)
- Black die shows any number except 2 (5 options) and white die shows 5
The total number of favorable outcomes is 5 + 5 = 10. Therefore, the probability is 10/36, which simplifies to 5/18.