Independent Events

Independent events do not influence the probability of another event. It is not to imply that an independent event or event’s outcome must be completely unrelated to a particular other feature of the situation, what-if question, or person and so forth; it just means that the outcome of an independent event cannot influence the probability of any other outcomes. For example, the probability that an event X will occur is simply the probability of an outcome of X divided by the probability of an outcome of any other event not involving X (“P(X)” divided by “P(not-X)”).

Events In probability:

An event is a collection of outcomes; in other words, an event is a subset of your sample space. It is important to note that an event has no probability by itself. An event will also not have any outcome that occurs with zero probability; 

For example, the outcome “rolling a 2 on Game dice” does not occur 0 times but rather can occur with the probability {P=0}.

Independent Events In Probability:

In probability, an independent event is an event whose outcomes are not dependent on one another.

Example: The probability of getting the red card from a standard deck of 52 playing cards consists only of the chances of drawing a specific suit from the deck, which is 25%, or 14. Assuming that no cards are removed or added to the deck after being shuffled, in which suits do we have an equal probability of drawing any one card: those with hearts, spades and diamonds or clubs? The answer is “none” for both cases. The probability of drawing a specific suit is not independent of the other suits and, therefore, cannot influence the probability of drawing any other suit. The probability that you will draw cards of all four suits is 1414 = 25%, and the probability that you will draw no cards at all is also 1414 = 25%.

By extending the example mentioned above, we can show that a set of events’ outcomes are not independent if they are not mutually exclusive(meaning they might occur or not).

The probability of drawing out a red card is not independent of the other suit because if you draw hearts, then it is likely that you will draw another card with hearts, which might lead to drawing another heart. Thus getting a red card is not an independent event.

Independent events are crucial in understanding probability because they can be assigned probabilities using the relative frequency of their outcomes. For example, the probability that an event A will occur is simply the probability of an outcome of A divided by the probability of an outcome of any other event not involving A (“P(A)” divided by “P(not-A)”). 

Another way to consider independent events is as a random variable X with a set of outcomes {x} and probabilities “P(X)” where “P(X)” = 1/the number of possible outcomes in X.

Independent events often directly relate to other statistical variables, like time or size in space. The word “independent” refers to variables that have this trait, but this usage is occasionally ambiguous. The words “independent” and “dependent” refer to a relationship between one variable and another. Dependent means that one variable changes or depends on the other. Independent variables are uncorrelated, meaning they do not depend on one another or change together. 

Examples of Independent Events:

These are some examples of situations where the events are independent and do not depend on each other for the outcomes.

1. You are flipping two coins. The outcomes are {HH, HT, TH, TT} with probabilities {0.5} for each type of outcome; what is the probability that HT occurs?

Ans. P(HH or HT) = P(HH) + P(HT) = 1 + 0.5 = 1.5

P(HH or TH) = P(HH) + P(TH) = 2.5 + 0.5 = 2.5

P(HT or TH)= 3

2. You have a jar containing 8 red marbles and 10 blue marbles; what is the probability you randomly pull out a red marble and it is shiny?

Ans. P(event HH, red marble shiny) = P(HH) + P(red marble shiny || HH) = 1 + 0.5

3. You possess two red and three blue marbles; you randomly draw a marble from the jar; what is the probability of a red marble?

Ans. P(red| red & blue) = P(red) * P(red|blue)

Conclusion: 

Independent events are not mutually exclusive, meaning their probability can be greater than the sum of their probabilities. This happens when two independent events occur simultaneously with each other. An example of mutually exclusive events would be rolling two dice: if one die comes up a 6, then the other die must not land on a 6, as they are mutually exclusive events. Independent events are crucial in understanding probability using the relative frequency of their outcomes. Independent events can predict the random occurrence of new, independent post-events, which means that the pre-event is independent of the post-event.