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When talking about probability, a sample space is a crucial concept. It includes all the possible outcomes of a particular experiment. For instance, consider the simple example of tossing a fair coin. In this case, the sample space consists of just two outcomes: heads (H) and tails (T). That's the most elementary version.

Now, let's make it more interesting by tossing the coin multiple times. Suppose we toss the coin four times. To determine the sample space, we list every possible sequence of these outcomes. For four coin tosses, each toss is like flipping a coin anew. Each flip is independent, meaning the result of one flip doesn't affect another.

To figure out the total number of outcomes for four tosses, you can use the formula for independent events. Since each toss has two possible results (H or T), and there are four tosses, the total number of outcomes is calculated as combinations of these possibilities: \(2^4 = 16\).

• The sample space for four coin tosses is: HHHH, HHHT, HHTH, HHTT, HTHH, HTHT, HTTH, HTTT, THHH, THHT, THTH, THTT, TTHH, TTHT, TTTH, TTTT.

This is a comprehensive list of all possible outcomes. Understanding this is vital for calculating probabilities.

A coin toss is a classic example in probability. It's simple yet effective in explaining fundamental concepts. When you toss a fair coin, it has two sides: heads (H) and tails (T). This makes it a binary option, lending itself well to clear outcomes. The fascinating part is that a single coin toss is a basic building block for more complex probability scenarios.

The nature of the coin being 'fair' ensures that the likelihood of it landing on heads is equally probable to the likelihood of it landing on tails. Thus, in a single toss, the outcome is independent and yields a 50/50 chance, or \( \frac{1}{2} \), for each side.

• One toss = two outcomes (H or T)
• Two tosses = four outcomes (HH, HT, TH, TT)
• Four tosses = 16 outcomes (as previously detailed in the sample space)

These figures grow exponentially, exemplifying how quickly combinations increase with more tosses. This simplistic principle of a coin toss serves as the foundation for more elaborate and varied experiments in probability, teaching learners how to analyze situations and assess possible outcomes.

In probability, equally likely outcomes are outcomes that have the same chance of occurring. This is a key aspect when dealing with fair experiments, like tossing a fair coin. Each potential outcome for a coin flip — getting heads or tails — is equally probable, at 50% or \( \frac{1}{2} \). This concept of equal likelihood carries across each independent toss.

Even when extending the experiment to multiple tosses, each sequence becomes a composite of equally likely single events. That's why in a sequence of four coin tosses, each of the 16 outcomes within the sample space has the same chance of occurring. Just as each coin toss is fair, each potential sequence of events is fair as well. This is calculated using the formula:

Probability of a specific outcome = \( \frac{1}{\text{total number of outcomes}} \)

Given that there are 16 possible sequences when tossing a coin four times, the probability of any specific one of these sequences occurring (like HTHH or TTTT) remains at \( \frac{1}{16} \).

• Every outcome holds an equal share in the probability.
• This is a fundamental tenet for making fair judgments in probabilistic scenarios.

Understanding equally likely outcomes ensures fair analysis and prediction in both simple and complex probability tasks.