Final answer:
To find the sum of all possible values of N, the probabilities for both balls being red or blue are calculated and set equal to the given total probability of 11/21. After solving, the possible values for N are found to be 4 or 16, hence the sum is 20.
Step-by-step explanation:
The student is asked to find the sum of all possible values of N, given that the probability of selecting two balls of the same color from a box containing 5 red balls and N blue balls is 11/21.
Calculating the Probability
There are two scenarios where the two balls selected can be of the same color: both are red or both are blue. We start by calculating the probability for each scenario separately.
For both balls being red, the probability is:
P(Red and Red) = (5/5+N) * (4/4+N)
For both balls being blue, the probability is:
P(Blue and Blue) = (N/5+N) * ((N-1)/4+N)
Since the total probability of selecting two balls of the same color is given to be 11/21, we set up the equation:
(5/5+N) * (4/4+N) + (N/5+N) * ((N-1)/4+N) = 11/21
After solving for N, we find that N can be 4 or 16. Therefore, the sum of all possible values of N is 4 + 16 = 20.