Problem 1 Two fair 6-sided dice are tossed, and the up face on each die is recorded. Find the probability of observing: A 5 appears on exactly one of the dice
Solution:
Let's call the dice A and B
If A = 5, then B could range from 1 to 6.
If B = 5, then A could range from 1 to 6.
There are 6+6 = 12 cases where either cube shows "5".
We need to subtract off 1 of those cases which is when A = 5 and B = 5 together.
There are 12-1 = 11 cases where exactly one of the dice shows "5".
This is out of 6*6 = 36 total dice rolls
Answer: 11/36
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Problem 2 Two fair 6-sided dice are tossed, and the up face on each die is recorded. Find the probability of observing: The sum of the numbers is 10 or more
Solution:
Here are the ways to achieve a sum of 10
4+6 = 10
5+5 = 10
6+4 = 10
Here are the ways to get a sum of 11
5+6 = 11
6+5 = 11
And finally we have 6+6 = 12 as the only way to get a sum of 12.
There are 3+2+1 = 6 ways to get what we want out of 36 total.
6/36 = 1/6
Answer: 1/6
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Problem 3 Two fair 6-sided dice are tossed, and the up face on each die is recorded. Find the probability of observing: The difference of the numbers is 2 or less
Solution:
Here is the sample space where we get a difference of 2.
3-1 = 2
4-2 = 2
5-3 = 2
6-4 = 2
and here's the flip of that
1-3 = -2
2-4 = -2
3-5 = -2
4-6 = -2
If we apply absolute value, then each of the negative results turn positive.
We have 4+4 = 8 ways of getting a difference of 2.
Here is the sample space where we get a difference of 1 or -1
2-1 = 1
3-2 = 1
4-3 = 1
5-4 = 1
6-5 = 1
and
1-2 = -1
2-3 = -1
3-4 = -1
4-5 = -1
5-6 = -1
Use absolute value to make each negative result turn positive.
There are 5+5 = 10 ways to get a dice roll such that we get a difference of 1.
Then we have to consider doubles in which we get a difference of 0
Example: 5-5 = 0
There are 6 instances of doubles, which means there are 6 instances that we get a difference of zero.
To summarize:
8 instances of getting a difference of 2.
10 instances of getting a difference of 1.
6 instances of getting a difference of 0.
There are 8+10+6 = 24 ways to get a difference of 2 or less (i.e. a difference of 2, 1, or 0)
24 ways to get what we want out of 36 total
24/36 = (2*12)/(3*12) = 2/3
Answer: 2/3