I am lost. How do I figure this out? I know there are 2 ways to get the sum of 5 but do I count it backwards too? 1+4 4+1. How do I know how many sums?
Joe rolled two fair number cubes that have faces labeled 1 to 6 each.
He rolled the two cubes 180 times. According to the theoretical probability,
how many of the rolls should Joe expect the sum to be 5?
-->1 2 3 4 5 6
1 ----- *
2 ----*
3 --*
4 *
5
6
36 possible outcomes
get 5 from four of them
so p = 4/36 = 1/9
(1/9)180 = 20 times
To figure out how many ways you can get a sum of 5 when rolling two numbered cubes, you need to consider all possible combinations of numbers on the cubes.
There are two ways to approach this problem:
1. Enumerate all the possible combinations:
- Start by listing all the possible outcomes for each of the cubes rolled. Since each cube has 6 faces labeled with numbers from 1 to 6, there are a total of 6 * 6 = 36 possible outcomes.
- Check each combination and count the ones that add up to 5. The combinations that result in a sum of 5 are: 1 + 4, 2 + 3, 3 + 2, and 4 + 1. So, there are four different combinations.
2. Use a systematic approach:
- To find the number of possible combinations that add up to 5, start by listing the possible outcomes for one cube and compare them to the sum you're looking for.
- Begin with one cube's outcomes: 1, 2, 3, 4, 5, and 6.
- For each number, calculate the corresponding number needed on the other cube to get a sum of 5. For example, to get a sum of 5 with a 1 on one cube, you need a 4 on the other cube.
- Calculate the number of combinations that result in a sum of 5 by counting the pairs. In this case, there are four combinations: (1, 4), (2, 3), (3, 2), and (4, 1).
Now, let's apply this to the given problem. The theoretical probability can be calculated by dividing the number of successful outcomes (sum of 5) by the total number of possible outcomes.
If you use the systematic approach mentioned above, you have found that there are four combinations that result in a sum of 5. Each combination has an equal probability of occurring, so the probability of rolling a sum of 5 is 4 out of 36.
To find out how many times Joe should expect to roll a sum of 5 out of 180 total rolls, you can use the proportion:
(Number of successful outcomes / Total number of possible outcomes) = (Number of expected successful outcomes / Total number of rolls)
Substituting the known values:
(4 / 36) = (Number of expected successful outcomes / 180)
Now cross-multiply the equation:
(4 * 180) = (36 * Number of expected successful outcomes)
Simplifying:
720 = 36 * Number of expected successful outcomes
Dividing both sides by 36:
Number of expected successful outcomes = 720 / 36 = 20
So, Joe should expect to roll a sum of 5 about 20 times out of 180 rolls, according to the theoretical probability.