Answer:
The probability that you will get two of the same colored gumballs WITHOUT reposition is 5/15,972 or 0.0003%, and the probability that you will get two of the same colored gumballs WITH reposition is 1/1,296 or 0.0008%
Step-by-step explanation:
Let's recall that the formula of the probability is:
Probability = Number of favorable outcomes/Total number of favorable outcomes
Total number of favorable events = 4 + 6 + 2 = 12
Reposition means that we take the gumball and put it back in the machine.
1. Therefore, the probability of getting two green gumballs without reposition is:
Probability of the 1st gumball to be green = 4/12
Probability of the 2nd gumball to be green = 3/11
Probability of getting two green gumballs without reposition = 4/12 * 3/11 = 12/132 = 3/33 = 1/11
The probability of getting two green gumballs with reposition is:
Probability of the 1st gumball to be green = 4/12
Probability of the 2nd gumball to be green = 4/12
Probability of getting two green gumballs with reposition = 4/12 * 4/12 = 16/144 = 1/9
2. Therefore, the probability of getting two red gumballs without reposition is:
Probability of the 1st gumball to be red = 6/12
Probability of the 2nd gumball to be red = 5/11
Probability of getting two red gumballs without reposition = 6/12 * 5/11 = 30/132 = 5/22
The probability of getting two red gumballs with reposition is:
Probability of the 1st gumball to be red = 6/12
Probability of the 2nd gumball to be red = 6/12
Probability of getting two red gumballs with reposition = 6/12 * 6/12 = 36/144 = 1/4
3. Therefore, the probability of getting two yellow gumballs without reposition is:
Probability of the 1st gumball to be yellow = 2/12
Probability of the 2nd gumball to be yellow = 1/11
Probability of getting two yellow gumballs without reposition = 2/12 * 1/11 = 2/132 = 1/66
The probability of getting two yellow gumballs with reposition is:
Probability of the 1st gumball to be yellow = 2/12
Probability of the 2nd gumball to be yellow = 2/12
Probability of getting two red gumballs with reposition = 2/12 * 2/12 = 4/144 = 1/36
Finally, the probability that you will get two of the same colored gumballs WITHOUT reposition is:
1/11 * 5/22 * 1/66 = 5/15.972 or 0.0003%
and the probability that you will get two of the same colored gumballs WITH reposition is:
1/9 * 1/4 * 1/36 = 1/1,296 or 0.0008%
