Source: Image of Dice, PD, bit.ly/1QRBI33; Image of Parking Meter, PD, http://bit.ly/1n2ZhxY;Image of coin, PD, http://bit.ly/1PLCvWB;Image of baseball, PD, http://bit.ly/1UuOQOa;Image of Capitol, PD, http://bit.ly/1mA8YUH;Image of cap, PD, http://www.clker.com/clipart-graduation-hat.html;Image of storm, PD, http://www.clker.com/clipart-10750.html;Image of snow, PD, http://www.clker.com/clipart-6490.html;Image of airplane, PD, http://www.clker.com/clipart-plane-7.html;Image of queen, PD, bit.ly/225X5Iu;Image of cards, PD,bit.ly/1P3XJcW
[MUSIC PLAYING] Hi, Dan Laub, here. And in this lesson, we're going to discuss identifying independent and dependent events. But before we get started, let's cover the objective for this lesson. By the end of this lesson, you should be able to know the difference between a dependent and an independent event. So let's get started.
Remember, from a previous lesson, that 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. Such a probability is represented using the terminology you see here, where the probability of an event, A, is equal to P of A, or P parentheses (A).
It is important to note that an event is simply a set of specific outcomes, as opposed to all possible outcomes. For example, think about rolling two dice like the ones you see pictured here. 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. So in this case, it is clear that the event does not consist of all possible outcomes.
Generally speaking, there are two kinds of events. One of these types of events 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. One example of 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. An example of two events that are dependent is parking in an illegal parking space and getting a parking ticket. In order to get the ticket, one must be parked illegally to begin with. 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. The reason we multiply the two probabilities is due to the fact that multiplication accounts for all the potential ways that two events can occur together. If one were to add the probabilities, they would not be accounting for all of these possible outcomes.
What this means is that if two different events-- let's call them 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 following term. Getting back to the coin toss example, the probability of tossing tails on one toss followed by heads on the next is equal to the probability of tossing tails multiplied by the probability of tossing heads, or this expression you see right here.
With regard to dependent events, mathematically speaking that is, two events are considered to be dependent, meaning that 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 you see here. For example, suppose one were to park illegally and risk getting a parking ticket. The probability of parking illegally and getting a ticket is not equal to the probability of parking illegally multiplied by the probability of getting a ticket, or this expression.
So let's consider two different examples here. Let's look at two different events. We'll call one, the price of gasoline increasing, and the second, a specific baseball team winning the World Series. The second example, event A, would be a college graduation rate, and event B would be how much rainfall Washington DC gets.
In both of these situations, the two events described, clearly are independent. One would certainly not expect the probability of gasoline prices rising and the probability of a baseball team winning the World Series to be connected. Nor would they realistically believe that the probability of a student graduating from college and the probability of rain in Washington DC on a given day are related either.
Considering the second example, suppose that the probability of an entering college freshman graduating in four years is equal to 0.25. Well, the probability of it raining in Washington DC on a Friday in May is equal to 0.40, or a 40% chance of rain. This being the case, if we were to multiply the probabilities of these two events, we would have the following. So the probability of graduating and raining would be equal to 0.25 times 0.40, or 0.10.
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.
Another example would be, say the amount of snow that a city like Denver receives and the amount of flight delays at the Denver Airport, or drawing two face cards from a deck of playing cards, consecutively. In both of these situations, the two events described are clearly dependent. One would certainly expect the probability of flight delays and a snowstorm at an airport to be connected. One would also plausibly believe that the probability of drawing a face card-- meaning a jack, queen, or a king-- from a deck of playing cards is related to the number of cards already drawn from the deck.
Considering the playing card example, the probability of drawing a jack, queen, or a king from a well-shuffled deck of playing cards is 12 in 52, or 0.23. Meaning, there are three face cards for each suit and four suits. So there are 12 possible face cards to be taken from a deck of 52 cards.
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. Why is it 11 out of 51? Well, that's relatively simple, because there are only 51 cards left, since we've already drawn one. And one of the face cards is already taken, meaning there are 11 of them left.
This being the case, if we were to multiply the probabilities of these two events, we would have the following. Well, the probability of drawing a face card, given that one has already been drawn, inequal to 0.23 times 0.216, or approximately 0.05. So since one face card has already been removed from the deck on the first draw, the probability of drawing a second face card is not equal to 12 out of 52 multiplied by 12 out of 52. Rather, the probability of drawing a second face card is now 11 out of 51 instead of 12 out of 52, 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. In this circumstance, these events are clearly dependent.
So let's go back to our objective, just to make sure we covered what we said we would. By the end of this lesson, we wanted to be able to know the difference between an independent and dependent event, which we did. We went over several examples of both.
So again my name is Dan Laub. And hopefully, you got some value from this lesson.
Two events are considered dependent if one event occurring does affect the probability that the other event occurs.
Two events are considered independent if one event occurring does not influence the probability of the other event occurring