Final answer:
Darren can win by drawing a white ball from a box that originally has 2 white and 4 red balls. The probability that he wins on his first turn is 2 out of 6. If he doesn't win immediately, different scenarios with reduced probabilities occur, as balls are drawn without replacement.
Step-by-step explanation:
The student is asking about the probability that Darren wins in a game of drawing balls from a box without replacement. Initially, the box contains four red balls and two white balls. Since Darren plays first, the probability that he wins by drawing a white ball on his first turn is 2 out of 6 (since there are 2 white and 4 red balls).
However, if he doesn't draw a white ball in the first turn (which happens with a probability of 4 out of 6), Marty then has a chance to draw from a box with three red and two white balls. If Marty also fails to draw a white ball, then the probability, when it's Darren's turn again, would be the probability of drawing one of the now one remaining white ball out of four total balls.
The calculation involves combining the probabilities for different scenarios in which Darren could win, considering that each draw without replacement changes the composition of the box for the next player's turn. The different scenarios are Darren winning on the first draw, or on subsequent draws after both players have failed to draw a white ball.