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Precalculus (2018 edition)
Independent & dependent probability
This time around we're not going to tell you whether we're working on a dependent or independent probability event problem. You tell us! Created by Sal Khan and Monterey Institute for Technology and Education.
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Does Probability and Statistic are from the same family? In their use I mean(38 votes)
Short version of my answer is Yes. If we were to categorize math concepts into a limited number of labels (say 20 labels), probability and statistics would be categorized in the same family. The reason lies on the fact that both of them are dealing with "uncertainty".
In probability, you have uncertain events in future. What you do is:
1- build a model (in example of flipping a coin, your probability model includes one variable with two possible values: H or T, and the variable will take on each of these two values with the same likelihood)
2- try to "predict the FUTURE" : meaning that what events will occur with what likelihood.
So, in a sense, you want to start from your "model" and reach some "data".
In statistics, you're still dealing with uncertainty. Because you don't have access to the whole population, only to a small subset of it (sample). what you do is:
1- You have the data. Using statistical tools, you try to build the best probability model that describes your data.
2- With that model in hand, you want to reason about the whole population, or in a sense, you want to "look backward" and identify the whole population that your sample is a part of.
So, you start from your "data", and you move toward a "model", that best describes your data.(69 votes)
Does anybody know what is mutually exclusive and independent event for venn diagram???(14 votes)
When the areas do not cross.(12 votes)
- In the case of independent events(A and B),
P(A|B) = P(A)
Could someone please explain the logic behind this?(6 votes)- Independent events have no effect on each other. If I have a deck of cards and a coin, the probability I draw a heart out of the deck of cards is not influenced by whether I had flipped a “heads” or “tails” prior to drawing the card. Likewise, the probability I flip a “heads” on the coin is not influenced by whether or not I drew a heart out of the deck prior to flipping the coin. The events of flipping the coin and drawing a card are independent of each other.
If event A is getting a “heads” by flipping a coin and event B is drawing a heart out of a deck of cards. The probability of getting a “heads” P(A) is no different than the probability of getting a “heads” given I have drawn a heart out of the desk first P(A|B). They are both a 0.5 probability. P(A)= P(A|B) for independent events.
The “given” event in the P(A|B) should be treated as though it has already happened – even if the probability of the “given” event is extremely rare. The probability of getting a “heads” given that you won the lottery is no different than the probability of getting a “heads” given that you did not win the lottery is the same as the probability of getting a "heads" before you know the lottery results. I hope that makes sense. This is true because the results of the lottery have no effect on the results of the coin flip – they are independent events.(13 votes)
I'm very confused about interdependence. In many of the question's hints we see the logic:
"P(A | B)=P(A) and P(B | A)=P(B) therefore events A and B are independent".
...or similar arguments. But the results above depend on the distribution of the balls, or cards, or whatever is under evaluation. Surely we should say that if we pick a subset of all possible events, and then pick a second subset of events from the first subset, the answer is always dependent regardless of whether "P(A | B)=P(A)" or not?
From mathisfun "Example: taking colored marbles from a bag: as you take each marble there are less marbles left in the bag, so the probabilities change. We call those Dependent Events, because what happens depends on what happened before"
Or how about his. I challenge you to the following game. We both put $1 on the table. We have a bag with 2 blue and 2 red, 4 balls altogether in it. I take out 2 balls look at it but don't show it to you. And I guess the color of the next ball. If I'm right then I win the $2 otherwise I lose. Is that fair? No. If I pulled out a ball of each color, the odds are 50-50, but if I pulled out 2 balls of the same color then I'm guaranteed to win. In the long term I will win and therefore the events are no independent regardless of the distribution of the balls. Or am I missing something?(5 votes)
This difficulty (which is giving me headache as well) is in the contrived little stories the problem writers are trying to make up to humanize the issue and they're shooting us all in the foots. We're never going to pull marbles out of bags at a circus for money and very few of us will become professional card counters in Las Vegas or the French Riviera anyway so let's leave that behind.
Imagine we are the creators of the situation, a system, a little computer game. Or no wait, imagine we each already created our games and then we swapped, now we're examining a system a classmate created.
Here's what I have so far (just starting thinking this way so anyone ahead of me and doing well with this please chime in):
Two events or behaviors within the system can be seen to be independent if the probability of one of them happening is unaffected by changes made to the other. In shorthand code: Independent is when P(A|B)=P(A). In human words A is going to do whatever it does regardless of what B does. If one is causing or interfering with the other it's dependent and you can spy that out by noticing that P(A|B) CHANGES from P(A).
And so on.
Hope this helps.(3 votes)
- What is the difference between mutually exclusive and independent events?(3 votes)
Independent events don't have a link between their probabilities, they can't affect each other. Mutually exclusive means that if the first has probability p, the other must have it as (1-p).(7 votes)
What does independent and dependent mean?(2 votes)
An independent event is an event in which the outcome isn't affected by another event. A dependent event is affected by the outcome of a second event.
Using the example of the ticket drawing, the dependency is established in the second drawing, as with ticket A no longer in play, the possible outcomes were reduced to only tickets B and C.(7 votes)
- Is that true? P(A)*P(B)=P(A&B) where events A and B are independent like flipping a coin twice to get 2 heads? ..then how can we judge this:
if 5 of 7 students forgot lunch, then, P(choose 2 student neither forgot lunch) = (2/7)(1/6)?
these 2 events are not independent and despite that we multiply their probabilities of the 2 events to get the probability of both together.(2 votes)
The 1/6 is not the probablility that the second student doesn’t forget his lunch. Rather, this is the conditional probability that the second student does not forget his lunch given that the first student does not forget his lunch. (The unconditional probability that the second student doesn’t forget his lunch, with no knowledge of the outcome for the first student, is actually 2/7.)
Whether events A and B are independent or not, it is always true that P(A and B)=P(A)P(B given A) as long as P(A) is nonzero. In the special case that A and B are independent, P(B given A) simplifies to P(B) in this equation.(2 votes)
- as in football games there are larger number of tickets than just 3. doesn't that huge number (i guess it will be in tens of thousands) allow us to consider the two events as independent even we do not put the first pulled ticket back?
i think later in this series you mentioned something about that, is this applicable for the example in subject?(2 votes)
In statistics the rule of thumb is that we can assume independence as long as the sample size (in this case the number of tickets pulled) doesn't exceed 10% of the population size (the total number of tickets). This is because the change in probability won't be large enough to make any real difference.(2 votes)
- Can you think of dependent and independent like positive and negative. For example in the video clip the first event was independent however, because the second event was dependent then both must be classified as dependent, or the result is dependent. (Negative) If they were both independent or positive they would still be positive or independent. However, if you have two dependent events or negative, does that make it independent or positive>(1 vote)
- That's an interesting connection to draw, but nope, in a sequence of events they are either all independent or all dependent. You might think about it like this:
Does the result of any event depend on any other events? Or in other terms: Are we likely to get a different result based on some of the other results we've observed? If so, these are dependent events. All of them.(4 votes)
- What is the law of total probability?(1 vote)
There is a .003% chance of achieving that squad on FIFA. Considering that, there is a 89% chance that you will win every game while playing with this team(2 votes)
Video transcript
The marching band is holding a
raffle at a football game with two prizes. After the first ticket is pulled
out and the winner determined, the ticket is
taped to the prize. The next ticket is pulled out to
determine the winner of the second prize. Are the two events
independent? Explain. Now before we even think about
this exact case, let's think about what it means for events
to be independent. It means that the outcome of one
event doesn't affect the outcome of the other event. Now in this situation, the first
event-- after the first ticket is pulled out and the
winner determined-- the ticket is taped to the prize. Then the next ticket is pulled
out to determine the winner of the second prize. Now, the winner of the second
prize-- the possible winners, the possible outcomes for the
second prize, is dependent on who was pulled out for
the first prize. You can imagine if there's
three tickets, let's say there's tickets A, B,
and C in the bag. And for the first prize,
they pull out ticket A. That's for the first prize. Now, when we think about who
could be pulled out for the second prize, it's only going
to be tickets B or C. Now the first prize could
have gone the other way. It could have been
A, B, and C. The first prize could have
gone to ticket B. And then the possible outcomes
for the second prize would be A or C. So the possible outcomes for
the second event, for the second prize, are completely
dependent on what happened or what ticket was pulled out
for the first prize. So these are not independent
events. The second event-- the outcomes
for it, are dependent on what happened in
the first event. So they are not independent. The way that we could have
made them independent is, after the first ticket was
pulled out, if they just wrote down the name or something,
and then put that ticket back in. Instead they taped
it to the prize. But if they put that ticket
back in, then the second prize, it would have still had
all the tickets there. It wouldn't have mattered who
was picked out in the first time because their name was just
written down, but their ticket was put back in. And then you would have
been independent. So if you had replaced the
ticket, you would have been independent. But since they didn't replace
the ticket, they taped it to the prize, these are not
independent events.