An event consists of any collection of possible outcomes of an experiment, while a probability is a measurement of how probable it is that an event will occur. The probability of an event, A is equal to P of A, or P(A).
It is important to note that an event is simply a set of specific outcomes, as opposed to all possible outcomes. Think about rolling two dice. An event would be rolling doubles, where each die has the same number on it, and there are six possible outcomes that would allow that to happen. However, all possible outcomes would also include rolling combinations that were not doubles. In this case, the event does not consist of all possible outcomes.
Generally speaking, there are two kinds of events. One is not impacted by the occurrence of another event, while the other type is impacted. Such events can be categorized as independent and dependent events, respectively.
Two events are considered independent if one event occurring does not influence the probability of the other event occurring.
An independent event is a coin toss. The probability of tossing tails is not influenced by prior coin tosses, as the probability of a coin toss resulting in tails is equal to 0.5, regardless of the number of times the coin has been tossed before.
Two events are considered dependent if one event occurring does affect the probability that the other event occurs.
Two dependent events are parking in an illegal parking space and getting a parking ticket. To get the ticket, you must be parked illegally. So, parking illegally does influence the probability of getting a ticket.
Mathematically speaking, two events are considered independent of one another if the probability of both events occurring is equal to the probability of the first event occurring multiplied by the probability of the second event occurring. This is the formula for independent events.
The reason we multiply the two probabilities is because multiplication accounts for all the potential ways that two events can occur together. If we add the probabilities, they would not be accounting for all of these possible outcomes.
What this means is that if two different events, A and B, are independent, the probability that both event A and event B occur is equal to the probability of event A occurring multiplied by the probability of event B occurring. This can be expressed by the term:
With regard to dependent events, the probability that both event A and event B occur is unequal to the probability of event A occurring multiplied by the probability of event B occurring. This can be expressed by the term:
Take a look at the math. Suppose that the probability of a college freshman graduating in four years is equal to 0.25, and the probability of it raining in Washington, DC, on a Friday in May is equal to 0.40, or a 40% chance of rain. If you were to multiply the probabilities of these two events, you would have:
Of course, it seems very implausible that the weather in a specific location has anything at all to do with how long it takes a college freshman to graduate. Remember that, for independent events, the probability that both event A and event B occur is equal to the probability of event A occurring multiplied by the probability of event B occurring. In this circumstance, it seems entirely plausible that these events are independent.
Consider the playing card example. There are 12 possible face cards to be taken from a deck of 52 cards. The probability of drawing a jack, queen, or king from a well-shuffled deck of playing cards is 12 in 52, or 0.23.
The probability of drawing a second face card, assuming one has already been drawn on the first try, is now 11 in 51, or 0.216. If you were to multiply the probabilities of these two events, you would have 0.23 times 0.216, or approximately 0.05
The probability of drawing a face card, given that one has already been drawn, is not equal to 12/52 multiplied by 12/52. Rather, the probability of drawing a second face card is 11/ 51, since one card has already been taken.
Remember, that for dependent events, the probability that both event A and event B occur is not equal to the probability of event A occurring multiplied by the probability of event B occurring.
Source: This work is adapted from Sophia author Dan Laub
P(A and B) = P(A) P(B)