8. Probability

Probability is a measure of how likely an event is to occur. The probability of any event lies between 0 and 1, where: • 0 means the event is impossible, • 1 means the event is certain. Real-Life Examples: • Tossing a coin and getting heads. • Rolling a die and getting a 4. • Selecting a red ball from a bag of colored balls.

Probability quantifies uncertainty about outcomes of events. When we say something has a probability of 0, it means that event cannot happen at all; conversely, a probability of 1 indicates the event is certain to happen. Each event we observe in daily life—like flipping a coin or rolling a die—can be associated with a certain probability, which helps us make predictions based on those events.

Think of probability like the chances of winning a game. If you roll a die, there are 6 faces, and if you want to roll a 4, your chance (or probability) of that happening is 1 out of 6. This is similar to playing a game where you can either win or lose based on your chances!

1. Experiment: An action that produces outcomes. Example: Tossing a coin.
2. Trial: Each repetition of an experiment. Tossing a coin once is a trial.
3. Outcome: A possible result of a trial. For tossing a coin, outcomes are {Heads, Tails}.
4. Sample Space (S): The set of all possible outcomes. For a die: S = {1, 2, 3, 4, 5, 6}
5. Event: A subset of the sample space. Getting an even number = {2, 4, 6}

Understanding key terms in probability is essential. An experiment is simply an action to see what possible outcomes can occur. Each individual occurrence of this action is called a trial. For instance, when you toss a coin, the two possible results are known as outcomes. The sample space comprises all possible outcomes of that experiment, like all the numbers you can roll on a die. Finally, events are specific results we are interested in; for example, getting an even number when throwing a die.

Imagine you're baking cookies. The experimental action (experiment) is mixing the dough and baking it; each time you bake a batch is a trial. The outcome could be cookies that are either burnt or perfectly baked, making it crucial to define your desired event as 'perfectly baked cookies'.

If all outcomes of an experiment are equally likely, then the probability of an event is:

Number of favourable outcomes 𝑃(𝐸) = Total number of outcomes Example 1: A die is rolled. What is the probability of getting a number less than 5? • Favourable outcomes = {1, 2, 3, 4} → 4 outcomes • Total outcomes = 6 𝑃(< 5) = 4/6 = 2/3

Classical probability is used when all outcomes have the same chance of occurring. To find the probability of a specific event, you divide the number of ways that event can occur (favourable outcomes) by the total number of possible outcomes. In this example with a die, the outcomes of rolling a number less than 5 can occur four ways (1, 2, 3, or 4) out of a total of six possible outcomes, thus yielding a probability of 2/3.

Consider flipping a coin. Here, there are two potential outcomes - getting heads or tails. Each has one way of occurring, so the probability of getting heads is 1 out of 2, or 50%. This theoretical foundation helps us make predictions about outcomes in a variety of scenarios, from games to daily decisions.

• Sure Event: An event that always happens (Probability = 1) Example: Getting a number less than 7 when a die is rolled. • Impossible Event: An event that never happens (Probability = 0) Example: Getting a 9 on a standard die.

In probability, certain events will always happen, like rolling a number less than 7 on a standard die, which has numbers 1 through 6. This certainty gives it a probability of 1. On the other hand, there are events that cannot happen, such as rolling a 9 with a standard die. This impossibility makes the corresponding probability 0.

Think of it like a weather forecast. A forecast that guarantees rain tomorrow in a rainy season is like a certain event (probability 1), while predicting snow in the same conditions is impossible (probability 0) because it just can’t happen.

When the probability is based on actual experiments or observations. Number of times event E occurs 𝑃(𝐸) = Total number of trials Example 2: You toss a coin 100 times and get 52 heads. 𝑃(Head)= 52/100 = 0.52 Note: As the number of trials increases, experimental probability approaches theoretical probability.

Empirical probability is calculated from real-world experiments and reflects outcomes as they actually occur over time. By gathering a significant number of trials, the probability value you compute will tend to align more closely with theoretical probability. For instance, if you flip a coin 100 times and land on heads 52 times, you can say the empirical probability of heads is 0.52.

Imagine conducting a survey within your school to see how many students prefer chocolate ice cream over vanilla. If you ask 100 students and 55 prefer chocolate, the empirical probability that a randomly chosen student enjoys chocolate is 55 out of 100. This real-world data gathering gives you insight into student preferences.

Probability plays a major role in: • Machine Learning (e.g., predicting email spam). • Robotics (e.g., estimating object position). • Natural Language Processing (e.g., next-word prediction). • Medical Diagnosis (e.g., predicting disease based on symptoms). Example: Predicting Weather An AI system may say: "There is a 70% chance of rain tomorrow." This is a probabilistic prediction based on historical data and models.

In artificial intelligence, probability is essential for making informed decisions in uncertain environments. For example, machine learning algorithms often rely on probability to determine whether an email is spam or not based on prior data. Robotics uses probability to accurately estimate where objects are. Essentially, probability helps AI systems understand and infer outcomes based on historical patterns and data.

Consider your weather app. When it says there's a 70% chance of rain, it means that based on past weather patterns, rain has occurred 70 out of 100 similar days. This probabilistic information can help you decide whether to carry an umbrella or leave it at home.

If A is an event, then its complement (A') is the event that A does not occur. 𝑃(𝐴′)= 1− 𝑃(𝐴) Example: If the probability of winning a game is 0.3, then the probability of not winning is: 𝑃(not winning) = 1 −0.3 = 0.7

Complementary events are pairs of outcomes that cover all possible results. If you know the probability of one event occurring, you can easily calculate the probability of the opposite happening by subtracting from 1. In the case of winning a game with a probability of 0.3, the chance of losing or not winning (the complement) would simply be 1 minus 0.3, which gives you 0.7.

Think about a basketball player. If they have a shooting success rate of 30% (0.3), then there’s a 70% chance (1 - 0.3) that they will miss the shot. This ‘complement’ helps coaches strategize and understand the overall performance of their players.

• Probability is used in recommendation systems (e.g., YouTube, Netflix). • It helps AI make fair and balanced decisions. • Misusing probabilistic models can lead to bias or discrimination. For example, wrongly predicting a person is a threat based on flawed data. So, understanding probability is not just a math skill, but a foundation for ethical AI.

Probability impacts our daily lives significantly, particularly through AI systems that recommend content like movies and videos on YouTube or Netflix. However, ethical implications arise when these systems misuse probabilistic models, leading to biases against certain individuals or groups. It’s crucial for us to recognize that while probability is a mathematical concept, it also has moral implications in technology usage.

Imagine a hiring algorithm based on past employee data that suggests who to hire. If this algorithm favors certain demographics over others unjustifiably, it may discriminate against equally qualified candidates. This example illustrates how understanding probability is essential to ensure fairness in AI decision-making.

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