5.3. Probability

Probability measures the chance of occurrence of an event, expressed as a number between 0 and 1.

Probability is a way to quantify how likely or unlikely an event is to happen. It is measured on a scale from 0 to 1, where 0 means that the event cannot happen at all, and 1 means the event is certain to happen. For example, if we say there is a 0.5 probability of rain tomorrow, it means there is an even chance it might rain or not.

Think of a game of flipping a coin. There are two possible outcomes: heads or tails. The probability of landing on heads is 0.5, indicating there is an equal chance of it landing heads or tails. Just like estimating the chance of rain, we can easily understand probabilities through everyday events.

● Sample Space (S): The set of all possible outcomes. ● Event: A subset of the sample space representing outcomes of interest.

The sample space is a complete list of all possible outcomes that can occur in a given scenario. For example, in a dice roll, the sample space would be {1, 2, 3, 4, 5, 6}. An event is what we are interested in within this sample space. For example, the event of rolling an even number would be a subset of the sample space: {2, 4, 6}. Understanding sample spaces helps us figure out the total possibilities and narrow down events we are concerned about.

Imagine you're picking a snack from a bowl containing chips, fruits, and nuts. The sample space represents all the snacks available. If you're only interested in chips, your event is the subset of that sample space focusing only on the chips. It helps to visualize what your options are and what specifically you're after.

If all outcomes in a sample space are equally likely, the probability of an event E is given by: P(E) = Number of favorable outcomes / Total number of outcomes.

This definition lays the foundation for calculating probability when all outcomes are equally probable. To find the probability of an event E, you divide the number of ways that event can happen (favorable outcomes) by the total number of possible outcomes in the sample space. For example, if you want to find the probability of rolling a 3 on a dice, since there is one way to roll a 3 and six possible outcomes, the probability would be P(3) = 1/6.

Consider picking a ball from a bag that contains 3 red balls and 2 blue balls. If you want to find the probability of picking a red ball, you would use the formula: P(Red) = Number of red balls / Total number of balls = 3/5. This means there's a greater chance to pick a red ball compared to a blue one, just like having more candy options in a jar increases the chances of picking your favorite flavor.

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