Probability answers

1. I have two 10-sided dice, numbered 1-10. If I toss both of them, the sums are shown on this chart:

One place to start is to make a grid: (grey is numbers on the dice)

  1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10 11
2 3 4 5 6 7 8 9 10 11 12
3 4 5 6 7 8 9 10 11 12 13
4 5 6 7 8 9 10 11 12 13 14
5 6 7 8 9 10 11 12 13 14 15
6 7 8 9 10 11 12 13 14 15 16
7 8 9 10 11 12 13 14 15 16 17
8 9 10 11 12 13 14 15 16 17 18
9 10 11 12 13 14 15 16 17 18 19
10 11 12 13 14 15 16 17 18 19 20

(yes, I know this is a pretty big grid. I put it on the practice problems so that I could save the smaller grid problems for the quiz)

And mark evens, numbers over 10, equal to 10 and doubles (see my colored grids). Then count to find the probabilities

2. I have a 10-sided die and a 6-sided die. If I toss them both, what is the probability that I will get:

One place to start is to make a grid: (grey is numbers on the dice)

  1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10 11
2 3 4 5 6 7 8 9 10 11 12
3 4 5 6 7 8 9 10 11 12 13
4 5 6 7 8 9 10 11 12 13 14
5 6 7 8 9 10 11 12 13 14 15
6 7 8 9 10 11 12 13 14 15 16

Then I can color the grid to show each situation and count.

 

3. I have two funny spotted 6-sided dice. Die A has the numbers: 1, 1, 3, 3, 5, 6 and die B has the numbers 2, 2, 2, 4, 4, 4

We can make a grid here too; in this grid I am marking which die would be larger:

  1 1 3 3 5 6
2 B B A A A A
2 B B A A A A
2 B B A A A A
4 B B B B A A
4 B B B B A A
4 B B B B A A

4. A.

Josh: 2, 2, 2, 2, 6, 6
Sarah:3, 3, 3, 3, 3, 3

  2 2 2 2 6 6
3 S S S S J J
3 S S S S J J
3 S S S S J J
3 S S S S J J
3 S S S S J J
3 S S S S J J

Sarah has a higher number more often when playing against Josh

B.

Sarah:3, 3, 3, 3, 3, 3
Kelly: 4, 4, 4, 4, 0, 0

  4 4 4 4 0 0
3 K K K K S S
3 K K K K S S
3 K K K K S S
3 K K K K S S
3 K K K K S S
3 K K K K S S

Kelly wins more often when playing against Sarah

C.

Kelly: 4, 4, 4, 4, 0, 0
Brian: 1, 1, 1, 5, 5, 5

  4 4 4 4 0 0
1 K K K K B B
1 K K K K B B
1 K K K K B B
5 B B B B B B
5 B B B B B B
5 B B B B B B

Brian wins more often when playing against Kelly

D.

Brian: 1, 1, 1, 5, 5, 5
Josh: 2, 2, 2, 2, 6, 6

  2 2 2 2 6 6
1 J J J J J J
1 J J J J J J
1 J J J J J J
5 B B B B J J
5 B B B B J J
5 B B B B J J

Something really unusual is going on in #4

Notice that Josh wins against Brian, who wins against Kelly, who wins against Sarah, (so you'd thing Josh's die was the best, and Sarah's was the worst except...) and Sarah wins against Josh. It kind of reminds me of an MC Escher drawing or maybe this one.