# Name Algebra 1B Activities and Homework Name Algebra 1B activities and homework

May 15, 2009Probability: sums of dice page 1

## Probability: sums of dice

Today’s probability problems concern rolling dice and adding up the numbers. (If you’ve ever played a game such as Monopoly, you probably already have some awareness that certain dicesums are more likely than others.) As before, we will take two different approaches to investigating such probabilities:

• Approximate probability from a simulation: One way to study probabilities in a random situation is to conduct a simulation. This means: perform many repeated trials of the random action (in this case, roll dice many times), tallythe results, and use these results to estimate the approximate values of the probabilities.
• Theoretical probability from a list of outcomes: Another approach is to make a list of all the possible outcomes (allthe possible dice rolls), and use this list to find probabilities.

There’s an important difference between these two approaches. A simulation gives you an approximate estimated value of a probability. (If you conducted a simulation more than once, you’d probably get slightly different answers each time.) The list-of-outcomes approach gives you the exact theoretical value of the probability, and everyone should always get the same answer.

For example, if you rolled a die 1000 times, and a “5” came up 165 times, you could estimate the probability of rolling a “5” as= 0.165. The exact theoretical value of this probability is different, but only slightly:≈ 0.167. The simulation provides a fairly close approximation of the actual value, so a simulation can be a useful tool for estimating probabilities.

### Simulation: sum of two dice

Directions: Working with a partner, repeatedly roll a pair of dice and add the numbers. Tally thenumber of times that you get eachnumber as the sum. Then we will find approximate (estimated) probabilities using combined data from the whole class.

sum of two dice / your group’s tally / whole class total / approximate probability
based on whole class total
2
3
4
5
6
7
8
9
10
11
12

### Theoretical probability: sum of two dice

Step 1: Make a list of all the possible dice rolls and their sums. The list has been started for you.

1 + 1 = 22 + 1 = 3
1 + 2 = 32 + 2 = 4

Step 2: How many addition statements does the above list contain?

Step 3: Use your work above to complete this table of probabilities.

sum of two dice / exact theoretical probability
as a fraction / exact theoretical probability
as a decimal
2 / 1/36 / 0.028
3
4
5
6
7
8
9
10
11
12

### Compare results: sum of two dice

Copy your decimal probabilities results from the previous two pages into this table.

sum of two dice / estimated probability
found by simulation / exact theoretical probability
found from outcome list
2
3
4
5
6
7
8
9
10
11
12

Now answer the following questions.

1.Which dice sums had an estimated probability from the simulation that was greater than the exact theoretical probability?

2.Which dice sums had an estimated probability from the simulation that was less than the exact theoretical probability?

3.Find the dice sum where the estimated and theoretical probabilities are furthest apart. Howfar apart are they?

### Simulation: Sum of three dice

Directions: Now repeatedly roll three dice and add the numbers. Tally the number of times that you get eachnumber as the sum. Then we will calculate probability estimates using combined data from the whole class.

sum of three dice / your group’s tally / whole class total / estimated probability
based on whole class total
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

### Theoretical probability: Sum of three dice

What about making a list of all the possible sums of three dice: 1 + 1 + 1 = 3, 1 + 1 + 2 = 4, etc. The trouble is that the list would be very long. How long? That’s a counting problem, and the answer is 6 · 6 · 6 = 216. Therefore, it’s not very practical to make the whole list and use it to find probabilities. We need a more efficient approach. The next page leads you through a method of finding out how many times each dice sum would appear on the list of 216 dice rolls, without actually making the whole list.

The table below summarizes the results when two dice are added.

sum of two dice / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12
number of ways / 1 / 2 / 3 / 4 / 5 / 6 / 5 / 4 / 3 / 2 / 1

Now think about adding in the results of the third roll. Answer these questions.

1.Suppose that the third die shows 1. Add this number to the first two dice as summarized atthe top of the page, and complete the table below.

Third die
= 1 / sum of three dice / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13
number of ways

2.Now suppose that the third die shows 2. Add this number to the first two dice as summarized at the top of the page, and complete the table below.

Third die
= 2 / sum of three dice / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14
number of ways

3.Complete the table below to show the results for all the possible numbers on the third cube. Calculate totals to find the number of ways to get each sum.

Sum of all three dice
3 / 4 / 5 / 6 / 7 / 8 / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16 / 17 / 18
Third die = 1 / 1 / 2 / 3
Third die = 2 / 1 / 2
Third die = 3 / 1
Third die = 4
Third die = 5
Third die = 6
TOTALS / 1 / 3 / 6

4.All of the totals shown above should add up to 216. Check that this is true.
If not, try to correct your work above.

5.Using the totals from problem 3 on the previous page, complete this table of probabilities. Write each answer as a fraction and as a rounded decimal.

sum of three dice / exact theoretical probability
3 / 1/216 ≈ 0.005
4 / 3/216 ≈ 0.014
5
6
7
8
9
10
11
12
13
14
15
16
17
18

6.Use the table above to help you answer these other probability questions about rolling threedice and adding them.

a.What is the probability of rolling a 4 ora 14?

b.If 7 and 11 are considered “lucky numbers,” what is the probability of rolling a “lucky number”?

c.What is the probability of not rolling a 12?