Independent and dependent events and conditional probability

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14:13:16 of on-demand video • Updated December 2021

  • Visualizing data, including bar graphs, pie charts, venn diagrams, histograms, and dot plots
  • Analyzing data, including mean, median, and mode, plus range and IQR and box-and-whisker plots
  • Data distributions, including mean, variance, and standard deviation, and normal distributions and z-scores
  • Probability, including union vs. intersection and independent and dependent events and Bayes' theorem
  • Discrete random variables, including binomial, Bernoulli, Poisson, and geometric random variables
  • Sampling, including types of studies, bias, and sampling distribution of the sample mean or sample proportion, and confidence intervals
  • Hypothesis testing, including inferential statistics, significance level, type I and II errors, test statistics, and p-values
  • Regression, including scatterplots, correlation coefficient, the residual, coefficient of determination, RMSE, and chi-square
English [Auto] So far in this course, we've been talking about the probability of independent events. I want to take the opportunity in this video to look at the difference between independent and independent events and how we calculate probability or each of those kinds of events. So let's start with a quick review of independent events, which are simply events that don't affect one another. So, for example, flipping a coin. Every time I flip a coin, you can think about that as one trial or one experiment. Independent events are events where one trial does not affect the result of the next trial or the result of one experiment is not affected by the result of the previous experiment. Flipping a coin is a great example of that, because if I flip a coin one time trial No one. Says that I can either get on that trial heads or tails, I have two options. Now, let's say I want to flip the coin again. I'll have your trial, number two. So let's say in the first trial, I got heads, I got heads. I pick up the coin and I flip it again for trial number two. Well, in that case, I can either get heads or I can get tails if I got tails on the first flip and then I pick up the coin and I flip it again, I can either get heads or tails and my options, my outcomes are always going to be the same, no matter how many trials I run. Even if I flip the coin 20 times, every single time I flip the coin, I can either get heads or tails. Each flip is unaffected by any of the flips before it. So when I flip the coin for trial number two, I flip the coin the second time. It doesn't matter at all whether I got heads the first time or tails the first time. I've still got a fair flip and an equal chance of getting heads or tails. In other words, trial number two is completely unaffected by the result of trial. Number one, if I got heads on the first flip, it doesn't make it more likely that I'm going to get heads on the second flip. And it doesn't make it more likely that I'm going to get tails on the second flip. If I got tails on the first flip, I'm not more likely to get heads on the second flip or more likely to get tails on the second flip. I have an equal chance of getting heads or tails on the second flip, regardless of whether I got heads or tails on the first flip. So these are independent events because each trial doesn't have any effect whatsoever on any of the other trials. We see this illustrated mathematically when we use the multiplication rule for independent events. The multiplication rule lets us find the probability of getting a specific outcome on each of our trials or experiments, for example, if we wanted to say what's the probability that we get heads on the first flip and then heads on the second flip, the multiplication rule would allow us to do that because we're saying what's the probability that we get a specific outcome on trial number one? And then what's the probability that we get a specific outcome on trial number two? And what's the probability of those two things happening back to back? Well, remember, because the events are independent, they don't have any effect on each other. So the probability of Event A happening and event B happening on two separate trials. So event occurs on trial number one, event B occurs on trial. Number two is simply the probability of event A multiplied by the probability of event B. In other words, if I flip a coin two times and I want to know what's the probability that I get tails and then tails? Well, event A is tails. On the first flight. Event B is tails on the second flip. The probability of Event A. is what's the probability that we get tails on the first flip? Well, we know that that's one half because I have a 50 percent chance of getting either heads or tails on a single flip. What's the probability of event B. What's the probability that I get tails on the second flight? Well, remember, these are independent events. We use the multiplication rule with independent events. So I don't have to think about the first flip at all. Trial number two basically is I'm just flipping a coin. We can forget the trial. No one ever even happened. So I'm just flipping a coin. What's the probability that I get tails? Well, it's still one 1/2. So then the probability of getting tails and tails is going to be equal to one 1/2, multiplied by one half or one fourth. Or a 25 percent chance, and that should make sense to us, because we looked at the probability of getting to tails when we flip a coin two times or when we flip a coin two times, what are all of our possible outcomes? We can get heads and then heads. We can get heads and then tails. We can get tails and then heads or we can get tails and then tails. These are the only four possible outcomes that we can get when we flip a coin two times. And we wanted to know what's the probability of getting tails and tails? Well, there's four possible outcomes and only one that meets the criteria we're interested in. So the probability is one out of four or 25 percent. That's all the multiplication rule is telling us. It's giving us the probability that multiple independent events happen on consecutive trials. The other thing I want to say about independent events is that events are independent not because there's a 50 50 chance of them occurring. So we use the example here of flipping a coin. And when you flip a coin, you have an equal chance of getting heads or tails, a 50 percent chance of getting heads, a 50 percent chance of getting tails. Flipping a coin is not an independent event simply because these probabilities are equal. Flipping a coin will be an independent event because I can flip a coin once and then flip a coin again. And those two flips don't have any effect on each other. Another example might be a basketball player making free throws. So a professional basketball player might have an 80 percent chance of making any given free throw. The reason we know that is because this basketball player has taken, I don't know, 20000 free throw shots in his professional career. And over the course of that time, we've seen that he makes about 80 percent of those. So we say that his free throw percentage is 80 percent. We know that he makes about 80 percent of his free throws. So if that's the case, what is the probability that he makes two free throws in a row? We're going to assume those are independent events. So making the first free throw is going to have no effect on whether or not he makes the second free throw and vice versa. Each free throw is an independent event. So if we go back to our multiplication rule, since these events are independent and we say the probability of A and B. It's going to be equal to the probability of a multiplied by the probability of B. We're saying that. Event A is he makes the first free throw. Event B is he makes the second free throw. And then we want to say the probability of Event A. is 80 percent, he's an 80 percent free throw shooter. So we can say that the probability of Event A. is zero point eight, the probability of event B, what's the chance that he makes free throw number two? Well, he's an 80 percent free throw shooter, so he's going to have an 80 percent chance on every single shot. So, again, it's zero point eight. We could also express these as eight out of 10 or 80 percent. But either way, then we say the probability of a make and a make is equal to zero point eight times zero point eight. Which is going to be equal to zero point six four or 64 percent, so he has a 64 percent chance to make two free throws in a row. Now, sometimes that feels a little confusing when we think about a free throw shooter having an 80 percent free throw make percentage. Why would the probability that he makes two in a row be less than 80 percent? Why would it be 64 percent and not always just 80 percent? Well, we can think about it this way, sort of by taking it to an extreme. What's the probability of him making a million free throws in a row, even though he's an 80 percent free throw shooter, making one million free throws in a row is extremely unlikely. He's bound to miss one of those in there somewhere. I mean, he only makes eight out of ten. So two out of every ten shots he makes, he misses historically. So if he's going to take a million shots, he's going to miss some somewhere. So the more makes we want him to have in a row, the smaller percentage chance he has of doing it. In other words, he's less likely to make a million shots in a row than he is to make 1000 shots in a row. He's less likely to make 1000 shots in a row than he is to make ten shots in a row. The fewer number of shots we have, the more likely he is to make all of them. So even though he makes 80 percent of all of his free throws, asking him to make two in a row, he only has a 64 percent chance of doing that. And notice here that the probability of a success on each event was 80 percent versus the probability here flipping a coin. We called tails sort of a success, if you will. And the probability of that success was 50 percent. The probability of getting tails was 50 percent. The probability of making a free throw was 80 percent. So it's not the percentage that makes these independent events. It's the fact that free throw No one has no effect on free throw number two and vice versa. And the fact that coin flip No one has no effect on coin flip number two and vice versa. So those are independent events and how you apply the multiplication rule to find the probability that multiple independent events occur. Now, we want to contrast this with dependent events, so these were all independent events. And now we want to look at dependent events. Dependent events are events that do affect each other, and the most classic example is drawing cards from a deck of playing cards. So if you have a standard deck of 52 playing cards and you want to say something like, what's the probability that I pull a card from the deck and it's a king, and then I pull another card from the deck and it is also a king. In other words, pulling two kings from the deck in a row without replacing the first card, and that without replacing is the key to making these dependent events. So here's what we mean if we have poll number one. Or draw number one and pull number two. From the deck, and we are not replacing the cards. Well, the probability of getting a king on poll number one, there are four kings in a 52 card deck. So there are 52 possible outcomes, but only four that are, for me, a success. So the probability of pulling a king on poll number one is four out of 52. Now, let's say I don't put that first card back. I keep it in my hand. I want to pull another card from the deck, and I want that card to also be a king. Well, the probability of getting a king on the second poll is not four out of 52 because two things. There are no longer 52 cards in the deck. There are only 51 and there are no longer four kings. There are only three kings. So the probability of pulling a king on the second draw isn't four out of 52 now. It's three out of 51, in other words, what I pull out of the deck on the first pole is going to affect what I can pull out of the deck on the second pole. It depends whether or not I pulled a king on the first pole or I pulled a seven or I pulled some other card. I'm altering my options for pole number two. That's a great way to think about it. I'm altering my options for the second pole. My possibilities, my possible outcomes are not the same for pole number two as they were for pole number one. These are dependent events, events that affect one another, where the outcome of the first trial, the first experiment has an effect on the outcome of the second trial or the second experiment. Keep in mind, though, that if I'm pulling cards from a deck and I'm replacing the card each time, so for example, I pull one time, I get a card, then I put that card back into the deck. I shuffle the deck, put the deck on the table and pull a second card. I look at that, OK, I put that back into the deck, shuffle the deck again, put the deck down on the table, pull a third card. Those then are independent events. I'm replacing the card each time, reshuffling the deck. So my odds are going to be the same every time, because in each poll there I have a four 52 card deck and it's been randomly shuffled. So the likelihood of me pulling any one card is going to be the same every time. It's when I don't replace the cards and I just keep pulling out of the deck one after the other, that my odds for pulling any one card in particular are going to change between trials. So what probability formula do we use for dependent events? Well, it's the same as the multiplication rule in terms of we're looking for the probability of A and B. Here we had a formula for the probability that A and B both happen when we have independent events here will say the probability that A and B. Both happen, but for dependent events. Well, that probability is going to be the probability that A happens multiplied by this new notation we haven't seen before, which looks like this. And this means the probability that event B happens, this line means given so the probability that Event B happens, given that Event A has already happened when we have independent events, the probability that A and B both happen is just the product of the probability that it happens and the probability that B happens. But when it's dependent events, we look at the probability that A and B both happen as the probability that A happens on the first pole. You can kind of think of this as an independent event by itself because nothing has happened before. This is the first event. This is the first thing to happen. But then for the second event, we have to assume because A and B are dependent, we have to assume that Event A has already happened. So we can't just multiply by the probability of event B, we have to say the probability of event B, assuming that A has already happened. So, for instance, if we want those two kings in a row, then Event A would be a king on pole number one. Event B would be a king on. Poll number two, the probability that A happens well, again, when we have poll number one, we have a complete 52 card deck. There are four kings in the deck. So the probability that we poll a king is four out of 52. But the probability of event B, assuming that A has already happened, given that A has already happened, the probability of pulling a king on the second trial depends on the result of a assuming that we polled the king. The probability of pulling a king the second time is three because there are only three kings left in the deck, divided by 51 because there are only 51 cards left in total. Therefore, we can say the probability of King and King back to back is for over 52. The probability of event A multiplied by the probability of be given a or three over 51. We can actually reduce these fractions to lowest terms for over 52 is the same as one over 13 and three over 51 is the same as one over seventeen. So then one times one in the numerator gives us one 13 times 17 in the denominator gives us two hundred twenty one. So the probability of pulling two kings in a row is one in 221. In other words, there's less than a one percent chance of pulling a king two times in a row.