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A random phenomenon is an event or process that occurs unpredictably, often leading to several possible outcomes. In our case, the random phenomenon involves a basketball player taking four free throws.
Each shot's result is uncertain as it can either be a hit or a miss. This unpredictability is a key characteristic of random phenomena, compelling us to explore all potential outcomes.
Random phenomena are often observed in numerous real-world scenarios, such as coin tosses or dice rolls.
These events have elements of randomness, making them essential study subjects in probability and statistics.

The possible outcomes are the different results that can occur from a random phenomenon. For the basketball player shooting four free throws, these outcomes are most crucial.

• In the first scenario (Part a), the possible outcomes focus on each sequence of hits (H) and misses (M). Each throw is independent, so there are various sequences that can form, indicating whether each individual shot was a hit or a miss.
• In the second scenario (Part b), rather than tracking sequences, the emphasis is on the total number of successful shots. Therefore, the possible outcomes range from 0 to 4, showing how many shots in total were made successfully.

Understanding possible outcomes helps in forming a sample space, which is the collection of all these potential results.

Probability offers a way to quantify the likelihood of a particular outcome from a set of possible outcomes. In our basketball scenario, if each shot is equally likely to be a hit or miss, every sequence in our sample space for Part (a) has the same probability.
For Part (b), calculating the probability for each possible outcome (ranging from 0 to 4 successful shots) involves understanding combinations and how frequently those particular outcomes can occur.
Probability assigns a number, typically between 0 and 1, to these outcomes. A probability of 0 indicates an impossible event, while a probability of 1 means the event will definitely occur.
This helps in analyzing how likely or unlikely different outcomes are, based on past results or theoretical models.

Counting techniques are essential tools in probability and statistics for identifying the number of possible outcomes. These techniques help in constructing the sample space effectively, especially when dealing with multiple stages or steps in a random phenomenon, such as our basketball scenario.

• For Part (a), the counting technique involves determining sequences of hits and misses over four shots. Since each shot can independently result in either a hit or a miss, we multiply outcomes using the principle of multiplication. This yields a total of \(2^4 = 16\) possible sequences.
• For Part (b), the counting focuses on combinations, as we calculated the total number of successful shots. Here, we count how many ways each particular outcome (0 to 4 hits) can be achieved.

These techniques simplify complex problems by providing a structured approach to counting possible results, which is vital for establishing sample spaces and probabilities accurately.