To solve this problem, the key concept used here is that of a Geometric Distribution, since we are dealing with the probability of the number of trials needed to get a certain number of successes (in this case, pulling a brown candy out of the bag is considered a "success").
Given that the probability of pulling out a brown candy (p_brown) is 20% or 0.2, we want to find the probability of getting 4 or fewer brown candies.
For a geometric distribution, the cumulative distribution function (CDF) is used to compute the probability of getting a certain number of successes or fewer. In this case, we would be looking for the cumulative probability of getting 4 or fewer successes. This is defined as the sum of probabilities of getting 0, 1, 2, 3, or 4 successes.
By applying these parameters to the cumulative distribution function for a geometric distribution, we get the required probability - the probability of getting 4 or fewer successes, which is brown candies.
The calculated probability that we end up with is 0.5904.
Looking at the options given, the closest answer to our computed probability is 0.5904. Therefore, none of the options a) 0.7744 b) 0.2256 c) 0.7756 d) 0.2244 are correct according to the solution provided.