Probability Fundamentals
Probability measures the likelihood that particular events will occur. This section presents essential ideas such as probability experiments and the sample space, which define the framework for understanding how we may measure and predict outcomes.
Probability experiments are activities with measurable outcomes. A sample space, \(\boldsymbol{S}\) encompasses all potential results of an experiment.
Example
Consider the simple scenario where you flip a fair coin. What is the probability of getting heads?
This example shows that there is a 50% chance of flipping heads, which is a basic application of probability concepts.
Probability of an Event and Types of Probabilities
The probability of an event, represented as \(P(A)\), runs from 0 to 1, with 0 denoting impossibility and 1 indicating certainty.
Outlined below are the different probability types:
Empirical Probability: Based on observations or experimental outcomes. For example, if you flip a coin 100 times and get heads 55 times, the empirical probability of flipping heads is \( \cfrac{55}{100} = 0.55\)
- Theoretical Probability: Theoretical or classical probability, also known as a a priori probability, is derived from mathematical logic rather than empirical evidence. It is the type of probability based on the possible outcomes that are known in a situation where each outcome is equally likely.
- \(n(A)\) is the number of favorable outcomes
- \(n(S)\) is the total number of possible outcomes
- Subjective Probability: Based on intuition or personal judgement rather than precise calculation. For example, if you feel there's a high chance it will rain tomorrow because the sky is very cloudy, you might assign a subjective probability of 80% to the event of rain.
The formula for theoretical probability is expressed as:
Where:
Example
Consider a game where you have a bag of 10 marbles, 4 of which are red. If drawing a red marble is event \(A\), then the theoretical probability \(P(A)\) is calculated as follows:
\[ P(A) = \cfrac{4}{10} = 0.4 \]
This example demonstrates how theoretical probability is used to determine the likelihood of an event under idealized conditions.
Tree Diagram Usage
A tree diagram can further illustrate this by clearly showing all possible outcomes of an event, aiding in the visualization of how \( n(A) \) and \( n(S) \) are determined.
Put tree diagram here.
Example
Calculate the probability of drawing a red card from a standard deck of 52 playing cards.
First, we can identify the following values:
- # of Red Cards: \(n(A) = 26\)
- Total # Red Cards: \(n(S) = 52\)
We can then substitute the pertinent values into the Theoretical Probability formula and solve:
\(P(\text{Red Card}) = \cfrac{n(A)}{n(S)}\)
\(P(\text{Red Card}) = \cfrac{26}{52}\)
\(P(\text{Red Card}) = \cfrac{1}{2}\)
Therefore, we can determine the theoretical probability to determine the likelihood of drawing a red card is 50%.
Statistical Fluctuation and Theoretical Predictions
Statistical fluctuation refers to the variability that can occur in empirical probabilities when the number of trials is limited. This variability shows how real-world data might deviate from expected outcomes predicted by theoretical probabilities. For example, in a small sample of repeated coin flips, the actual ratio of heads to tails might significantly differ from the expected 1:1 ratio.
Theoretical predictions provide a baseline for expected outcomes under ideal conditions, assuming all outcomes are equally likely. These predictions are essential for hypothesis testing and validating the assumptions of probabilistic models, helping to understand the long-term behavior of probability experiments and smoothing out the short-term irregularities observed in empirical data.
Complement of an Event
The complement of an event, denoted \(A'\), represents the scenario where the event \(A\) does not occur. In probability theory, the sum of the probabilities of an event and its complement is always 1.
The Complement of an Event can be determined algebraically as such:
Where:
- \(P(A)\) is the probability of event \(A\)
- \(P(A)\) is the probability of the complement of event \(A\)
Consider a standard deck of 52 playing cards. A player randomly selects a card. Calculate the probability that the card drawn is not a club.
First, we can let event \(A\) be the event where a card is a club. Since there are 13 clubs in a deck of 52 cards, the event \(A\) corresponds to drawing one of these clubs.
Next, we can determine the probability of drawing a club by using the Theoretical Probability formula:
\(P(A) = \cfrac{n(A)}{n(S)}\)
\(P(A) = \cfrac{13}{52}\)
\(P(A) = \cfrac{1}{4}\)
The complement of event \( A \), denoted as \( A' \), is the event of drawing a card that is not a club. To find \( P(A') \), the probability of this complement, we use the following formula:
We can then substitute the pertinent values and solve:
\(P(A') = 1 - \cfrac{1}{4}\)
\(P(A') = \cfrac{3}{4} \)
Thus, there is a \(\cfrac{3}{4}\) or 75% chance that a randomly drawn card from a standard deck will not be a club.
First, we can let event \(A\) represent drawing a red marble. With 20 red marbles out of a total of 50, we can calculate the probability of drawing a red marble, \(P(A)\), as such:
\(P(A) = \cfrac{n(A)}{n(S)}\)
\(P(A) = \cfrac{20}{50}\)
\(P(A) = \cfrac{2}{5}\)
Next, we can calculate \(P(A')\) as such:
\(P(A') = 1 - P(A)\)
\(P(A') = 1 - \cfrac{2}{5}\)
\(P(A') = \cfrac{3}{5}\)
Therefore, we can determine there is a \(\cfrac{3}{5}\) or 60% chance that a randomly drawn marble from this bag will not be red.
Understanding Subjective Probability
Subjective probability measures the likelihood of an event based on personal judgment rather than objective data. It reflects an individual's educated guess about the outcome of an event, drawing on their experience and intuition.
Practical Application: This type of probability is commonly expressed in daily decisions and predictions, such as estimating the chances of success in an exam or predicting outcomes in sports and financial markets. For example, a student might feel 90% confident about passing an exam based on their preparation and previous performance.
Example
Estimate the probability that the next pair of shoes you purchase will be the same size as your previous pair.
While it's possible your foot size has slightly changed or that different shoe brands may fit differently, it's highly likely that your next pair of shoes will be the same size as your last. Given these factors, a subjective probability of 80% to 90% seems reasonable for the shoes being the same size as your previous pair.