Determine whether the following individual events are overlapping or non-overlapping. Then find the probability of the combined event.
Getting a sum of either 4, 9, or 10 on a roll of two dice
4 can be obtained by
2,2
3,1 or
1,3
9 can be obtained by
6,3
3,6
5,4 or
4,5
10 can be obtained by
6,4
4,6 or
5,5
For each, they are out of 36 possibilities.
To determine whether the events of getting a sum of 4, 9, or 10 on a roll of two dice are overlapping or non-overlapping, we need to analyze the possible outcomes.
First, let's determine the outcomes that result in a sum of 4:
- Rolling a 1 on the first die and a 3 on the second die: (1, 3)
- Rolling a 2 on the first die and a 2 on the second die: (2, 2)
- Rolling a 3 on the first die and a 1 on the second die: (3, 1)
Next, let's determine the outcomes that result in a sum of 9:
- Rolling a 3 on the first die and a 6 on the second die: (3, 6)
- Rolling a 4 on the first die and a 5 on the second die: (4, 5)
- Rolling a 5 on the first die and a 4 on the second die: (5, 4)
- Rolling a 6 on the first die and a 3 on the second die: (6, 3)
Finally, let's determine the outcomes that result in a sum of 10:
- Rolling a 4 on the first die and a 6 on the second die: (4, 6)
- Rolling a 5 on the first die and a 5 on the second die: (5, 5)
- Rolling a 6 on the first die and a 4 on the second die: (6, 4)
Now, let's see if any of these outcomes overlap. In this case, we can see that the outcomes (2, 2), (4, 5), (5, 4), and (5, 5) appear in more than one of the sums.
So, the events of getting a sum of 4, 9, or 10 on a roll of two dice are overlapping.
To find the probability of the combined event, we need to calculate the probability of each individual event and then add them up.
The probability of getting a sum of 4 is 3/36, or 1/12.
The probability of getting a sum of 9 is 4/36, or 1/9.
The probability of getting a sum of 10 is 3/36, or 1/12.
To find the probability of the combined event, we sum up these individual probabilities:
1/12 + 1/9 + 1/12 = 6/36 = 1/6.
Therefore, the probability of the combined event is 1/6.
To determine whether the individual events are overlapping or non-overlapping, let's analyze the outcomes of each event.
Event 1: Getting a sum of 4 on a roll of two dice.
The possible outcomes to get a sum of 4 are (1, 3), (2, 2), and (3, 1).
Event 2: Getting a sum of 9 on a roll of two dice.
The possible outcomes to get a sum of 9 are (3, 6), (4, 5), (5, 4), and (6, 3).
Event 3: Getting a sum of 10 on a roll of two dice.
The possible outcomes to get a sum of 10 are (4, 6), (5, 5), and (6, 4).
To find the probability of the combined event, we need to sum up the probabilities of each individual event happening.
P(Event 1) = Number of favorable outcomes / Total number of outcomes
P(Event 1) = 3 / 36 = 1 / 12
P(Event 2) = 4 / 36 = 1 / 9
P(Event 3) = 3 / 36 = 1 / 12
As the events have different outcomes, they are non-overlapping.
P(Combined event) = P(Event 1) + P(Event 2) + P(Event 3)
P(Combined event) = 1/12 + 1/9 + 1/12
P(Combined event) = 6/36 + 4/36 + 3/36
P(Combined event) = 13/36
Therefore, the probability of the combined event of getting a sum of either 4, 9, or 10 on a roll of two dice is 13/36.