91Ó°ÊÓ

The concept of a "sample space" is foundational in probability and statistics. It represents the set of all possible outcomes of a specific experiment. In our exercise, we are tossing three fair coins, where each coin has two possible outcomes: heads (H) or tails (T). Thus, each toss can be considered an independent event.

To find the full sample space for this experiment, we must list all possible sequences of heads and tails for the three coins. The sequences are composed as follows:

• HHH
• HHT
• HTH
• THH
• HTT
• THT
• TTH
• TTT

This gives us a total of 8 unique outcomes, reflecting the combinations possible with two outcomes per coin and three coins being tossed.

Each outcome is called a "sample point," and in this exercise, we assign a value of \(x\) to represent the number of heads in each outcome. This variable \(x\) enables us to quantify and analyze the results probabilistically.

A "probability histogram" is a type of graph that helps visualize the distribution of probabilities for a discrete random variable. It allows us to see at a glance which outcomes are more likely and how the probabilities are spread across different values.

In our exercise, the variable \(x\) represents the number of heads in three coin tosses. We have calculated the probability \(p(x)\) for each possible value of \(x\):

• \(x = 0\), where the outcome is TTT, has a probability \(p(0) = \frac{1}{8}\).
• \(x = 1\), where outcomes are HTT, THT, TTH, has a probability \(p(1) = \frac{3}{8}\).
• \(x = 2\), where outcomes are HHT, HTH, THH, has a probability \(p(2) = \frac{3}{8}\).
• \(x = 3\), where the outcome is HHH, has a probability \(p(3) = \frac{1}{8}\).

To construct the histogram, plot values \(x = 0, 1, 2, 3\) on the x-axis, and their corresponding probabilities on the y-axis. The height of each bar represents the probability, allowing you to visually compare the likelihood of each event. In this histogram, the bars for \(x=1\) and \(x=2\) are taller, indicating these outcomes are more probable.

A "binomial distribution" is a discrete probability distribution of the number of successes in a fixed number of independent experiments, each with the same probability of success. This type of distribution is characterized by two parameters: \(n\), the number of trials, and \(p\), the probability of success in each trial.

In the context of our exercise, each coin toss is an independent trial, and getting a head can be considered a "success." Here, \(n\) is 3, since we toss the coin three times, and \(p\) is 0.5, as the coin is fair with an equal chance for heads or tails.

The probability of achieving exactly \(x\) successes (heads in this case) out of \(n\) trials is given by the formula:\[P(x) = \binom{n}{x} p^x (1-p)^{n-x}\]

Using this, we can verify our earlier results of probabilities for each \(x\) value. For example, the probability of getting exactly two heads (\(x=2\)) is:\[P(x=2) = \binom{3}{2} (0.5)^2 (0.5)^{1} = 3 \times \frac{1}{4} \times \frac{1}{2} = \frac{3}{8}\]

Thus, a binomial distribution helps us understand the probability structure of our coin-tossing experiment, providing a straightforward way to calculate and represent these probabilities.