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Understanding the concept of sample space is a foundational aspect of combinatorics. In simple terms, the sample space refers to the collection of all possible outcomes of a random experiment. Considering April May's packet of gummy candies, her selection of four candies at random represents a sample space composed of different combinations of candies. This means each unique set of four candies constitutes an individual outcome within the sample space.

To effectively determine the sample space, one should consider all possible combinations of the candy flavors available. In this case, since there are four different flavors, and April May is picking four candies, the variety of combinations form the complete set of possible outcomes. It's important to count these outcomes to understand the probability of any specific event occurring. Calculating the sample space can also help one to visually understand the breadth of possibilities in stochastic scenarios, facilitating better comprehension of probabilistic events.

In scenarios involving random selection, calculating probabilities allows us to understand how likely an event is to occur. Probability is essentially a ratio that compares the number of ways a particular event can happen to the total number of possible outcomes in the sample space.

For April May's candy selection, once the sample space is set, calculating the probability of her selecting two strawberry and two black currant gums requires first identifying the specific combinations that meet this event criterion. Here, the probability is computed by taking the number of desired combinations (6 combinations, as determined earlier) and dividing by the total number of combinations possible (495). For this example, the probability of April picking this specific combination would be:

• The number of favorable outcomes = 6
• The total number of outcomes = 495
• Probability = favorable outcomes / total outcomes = \( \frac{6}{495} \)

This basic formula of dividing specific outcomes by the sample space is a standard method in probability calculations, helping to quantify the likelihood of specific scenarios within distinct sample spaces.

Combinations are the mathematical approach used to select items from a larger set without considering the order of selection. This concept helps in calculating the number of ways a selection can be made when the order doesn't matter. For example, selecting two strawberry and two black currant gummies out of the total types available, focuses solely on the count of each flavor chosen, not the order they are picked.

The formula used for combinations is denoted as \( C(n, k) \) or \( \binom{n}{k} \), which calculates the number of ways to choose \( k \) items from \( n \) total items. For April's choice of gummies, it helps to separately determine combinations for picking strawberry and black currant gums:

• Strawberry combinations: \( C(4, 2) = \frac{4!}{2!(4-2)!} = 6 \)
• Black currant combinations: \( C(2, 2) = \frac{2!}{2!(2-2)!} = 1 \)

The product of these gives the number of favorable combinations for this specific event. Mastering combinations allows students to approach a wide range of problems involving selection without replacement, forming an integral part of many probability and statistics courses.