# Events (Probability)

## The Fundamentals of Probability: Understanding Events and Their Probabilities

In the world of probability, an event is defined as an outcome or a group of outcomes that arise from an experiment. An experiment is a process that can be repeated multiple times and produces specific outcomes. The sample space is the collection of all possible results, making an event a subset of the sample space. For example, an event could be tossing a coin and getting a head or throwing a die and rolling a 6.

## The Likelihood of Events in Probability

The probability of an event ranges from 0 to 1 and measures the chances of that event occurring. A probability of 0 means the event is impossible, while a probability of 1 signifies that the event is certain to happen. If an event has a probability of 0.5, it is equally likely to occur or not occur. Any event with a probability between 0 and 0.5 is unlikely to happen, while any event with a probability between 0.5 and 1 is likely to occur. Let's delve deeper into this concept.

## Expressing Probabilities in Different Forms

Probabilities can be represented in fractions, decimals, or percentages. For instance, a probability of 1/2 is equivalent to 0.5 or 50%.

For example, let's say there is a bag with 6 red balls and 4 blue balls. If you randomly pick one ball without replacing it, what is the probability of getting a blue ball?

## Understanding Independent Events

Two events, A and B, are considered independent if the occurrence of A does not affect the probability of B happening, and vice versa. For example, when tossing a coin twice, the result of the first toss has no bearing on the second one. The probability of getting heads on the first toss is 1/2, and it remains the same for the second toss, regardless of the first outcome.

When two events are independent, the following multiplication rule applies:

**Probability of A and B occurring together = Probability of A x Probability of B**

For instance, if A and B are independent events where A = getting heads on a coin toss and B = rolling a 5 on a die, the probability of both events happening is 1/2 x 1/6 = 1/12.

## Investigating Dependent Events

Two events, A and B, are considered dependent if the occurrence of A affects the probability of B, and vice versa. For example, if you draw two cards from a deck without replacing the first card, the likelihood of getting an ace on the second draw will change depending on the first outcome.

If the first card was an ace, the probability of getting another ace on the second draw is 3/51 (since there are 3 remaining aces out of 51 cards). If the first card was not an ace, the probability of getting an ace on the second draw is 4/51 (since there are 4 aces remaining out of 51 cards).

The multiplication rule for dependent events is as follows:

**Probability of A and B occurring together = Probability of A x Probability of B after A has occurred**

## Understanding Mutually Exclusive Events

Mutually exclusive events have no outcomes in common, meaning they cannot occur at the same time. For example, getting heads or tails when tossing a coin are mutually exclusive events, since getting one outcome eliminates the possibility of the other.

This concept can be represented using a Venn diagram, as shown below:

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