Final answer:
To find the number of possibilities in various situations, factorial is used for arrangements, the number of possibilities raised to the power of repetitions for independent repeated events, and the ratio of successful to total outcomes for calculating probabilities.
Step-by-step explanation:
Finding the Number of Possibilities
To find the number of possibilities for various scenarios, different mathematical approaches are used. For permutations, such as arranging 4 unique items, we use factorial, which in the case of 4 items is 4! or 4 x 3 x 2 x 1, resulting in 24 combinations. This method can be practiced by systematically listing all 24 combinations to understand the ordering process.
When considering the number of possible outcomes in repeated independent events, such as drawing cards or rolling dice, we raise the number of possibilities for one event to the power of the number of repetitions. For instance, choosing a card from a deck (52 possibilities) five times without replacement would be 52^5 possibilities.
To calculate the probability of an event, divide the number of successful outcomes by the total number of possible outcomes. This can be illustrated with a two-event situation, like drawing two specific marbles from a bag. If we have a systematic list of outcomes and know there are 8 ways to draw blue first and 3 ways to draw red second, we have 24 possible combinations yielding the event we want.
Another approach is using a tree diagram to list out all the combinations, assess the theoretical probability, or even use a table to calculate probabilities of more complex events, such as tossing coins. Each approach relies on counting the successful outcomes and comparing it to the total possible outcomes.