91Ó°ÊÓ

Probability is a mathematical concept that measures the likelihood of an event happening. It is expressed as a number between 0 and 1 (or 0% to 100%), where 0 indicates an impossible event, and 1 indicates a certain event.

When tossing a coin, there are two possible outcomes: heads (H) or tails (T). The probability of getting heads or tails with a balanced coin is equal, which means the event of obtaining heads has a probability of 0.5, as does the event of obtaining tails. This scenario where each outcome has an equal chance of occurring is an example of a uniform probability distribution.

In a sample space, which is the set of all possible outcomes, we list each individual outcome once. For a single coin toss, the sample space is \( S = \{H, T\} \). When tossing a coin twice, we use the fundamental principle of counting to determine all possible combinations, leading to a sample space of \( S = \{HH, HT, TH, TT\} \).

A balanced coin is an idealized coin that has an equal chance of landing on heads or tails when tossed. This means that it has no physical bias that would make it more likely to land on one side over the other. Its weight is distributed evenly, and its shape is symmetric.

In terms of probability, the likelihood or chance of a balanced coin landing on heads is exactly the same as landing on tails, which is \( P(\text{Heads}) = P(\text{Tails}) = 0.5 \). Given this symmetrical property, when conducting probability experiments or exercises, a balanced coin provides a clear model of randomness and fairness.

What Does This Mean for Sample Spaces?

For a balanced coin tossed twice, as mentioned in the exercise solution, the sample space remains uncomplicated: \( \{HH, HT, TH, TT\} \). Each of these sequences is equally likely to occur.

In contrast to a balanced coin, a biased coin has a predisposition towards one of the outcomes. This could be due to its mass being unevenly distributed, or its shape being non-symmetrical. Consequently, the probabilities of landing on heads or tails when tossed are not equal.

The exercise provided a biased coin where heads are favored with a ratio of 3:1; this translates to probabilities of \( P(\text{Heads}) = 0.75 \) and \( P(\text{Tails}) = 0.25 \). However, this bias affects only the probability of each individual outcome, not the sample space itself.

Sample Space and Probability with Biased Coins

The sample space for a single toss of a biased coin is still \( S = \{H, T\} \), and for two consecutive tosses, it remains \( S = \{HH, HT, TH, TT\} \). But when calculating the probability of a sequence of outcomes from multiple tosses, we account for the bias, thus affecting calculations that depend on understanding and using the correct probabilities.