There are 9 counters in a bag. There is a number on each counter.

1, 1, 2, 2, 2, 3, 3, 3, 3.

Joanne adds together the numbers on the 3 counters to get her total.

What is the probability that he total is 6?

Thank you.

To find the probability that the total is 6, we need to determine the number of ways in which the counters can be selected such that their sum is 6, and then divide that by the total number of possible selections.

First, let's list all the possible combinations that can give a total of 6:

1 + 2 + 3 = 6
1 + 3 + 2 = 6
2 + 1 + 3 = 6
2 + 3 + 1 = 6
3 + 1 + 2 = 6
3 + 2 + 1 = 6

Next, let's count the number of ways we can arrange the numbers 1, 2, and 3. We can use the concept of permutations to find this.

The number of permutations of n different items taken r at a time is given by nPr = n! / (n-r)!.

In this case, we have 3 different numbers (1, 2, and 3), and we want to select all 3 of them. So, we can calculate the number of permutations as follows:

3P3 = 3! / (3-3)! = 3! / 0! = 3! = 3 * 2 * 1 = 6

So, there are 6 ways to arrange the numbers 1, 2, and 3.

Therefore, the probability that the total is 6 is given by the number of favorable outcomes (6) divided by the total number of possible outcomes.

Total possible outcomes: Since we have 9 counters in total, we need to calculate the number of combinations of 3 counters that we can select from the bag. This can be calculated using the permutation formula as follows:

9P3 = 9! / (9-3)! = 9! / 6! = 9 * 8 * 7 = 504

So, the total number of possible outcomes is 504.

Therefore, the probability that the total is 6 is:

Probability = (Number of favorable outcomes) / (Total number of possible outcomes) = 6 / 504 = 1/84.

So, the probability that the total is 6 is 1/84 or approximately 0.0119.

To find the probability that Joanne's total is 6, we need to calculate the number of favorable outcomes divided by the total number of possible outcomes.

Let's break it down step by step:

Step 1: Determine the total number of counters in the bag.
In this case, there are 9 counters.

Step 2: Identify the possible outcomes.
The possible outcomes are the different combinations of counters that Joanne could select from the bag to achieve a total of 6.

Step 3: Calculate the number of favorable outcomes.
To get a total of 6, Joanne needs to select three counters whose numbers add up to 6. The possible combinations to achieve this are:
- 1,2,3
- 1,3,2
- 2,1,3
- 2,3,1
- 3,1,2
- 3,2,1

So, there are 6 favorable outcomes.

Step 4: Calculate the total number of possible outcomes.
Since Joanne is selecting three counters from a bag of 9, the total number of possible outcomes can be calculated using the combination formula. The formula for combinations is: nCr = n! / (r!(n-r)!), where n is the total number of items and r is the number of items being selected at a time. In this case, n = 9 and r = 3.

Using the combination formula:
9C3 = 9! / (3!(9-3)!) = (9 * 8 * 7) / (3 * 2 * 1) = 84

So, there are 84 possible outcomes.

Step 5: Calculate the probability.
The probability of Joanne's total being 6 is given by:
(Number of favorable outcomes) / (Total number of possible outcomes) = 6/84 = 1/14 ≈ 0.0714

Therefore, the probability that Joanne's total is 6 is approximately 0.0714 or 7.14%.

150/504