A bag contains seven red balls and four yellow balls. If we select two balls at random, what is the probability that both balls will be yellow if this sample is without replacement?

To solve this problem, we need to use the concept of probability.

Step 1: Identify the total number of balls. We have 7 red balls and 4 yellow balls, so the total number of balls is 7 + 4 = 11.

Step 2: Calculate the probability of drawing the first yellow ball. The probability of an event is the number of ways that event can occur divided by the total number of outcomes. So, the probability of drawing a yellow ball first is 4 (the number of yellow balls) divided by 11 (the total number of balls), which is 4/11.

Step 3: Calculate the probability of drawing the second yellow ball. After we've drawn the first yellow ball, there are now 3 yellow balls left and a total of 10 balls left. So, the probability of drawing a second yellow ball is 3/10.

Step 4: Since we're looking for the probability of both events happening (drawing a yellow ball and then drawing another yellow ball), we multiply the probabilities from steps 2 and 3 together. So, the probability of drawing two yellow balls in a row is (4/11) * (3/10) = 12/110 = 0.1091 or approximately 10.91%.

So, the probability that both balls will be yellow if this sample is without replacement is approximately 10.91%.