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Instructor: Hey everyone.
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Life is filled with uncertain events and often
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we must consider the possible outcomes before deciding.
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We ask ourselves questions like
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What is the chance of success
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and what is the probability that we fail
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to determine whether the risk is worth taking.
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Many CEOs need to make huge decisions when investing
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in their research and development departments
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or contemplating buyouts or mergers.
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By using probability and statistical data
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they can predict how likely each outcome is
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and make the right call for their firm.
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Some of you might be wondering
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what is this probability we're talking about?
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Essentially probability is the
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chance of something happening.
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A more academic definition
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for this would be the likelihood of an event occurring.
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The word event has a specific meaning
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when talking about probabilities.
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Simply put, an event is a specific outcome
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or a combination of several outcomes.
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These outcomes can be pretty much anything.
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Getting heads when flipping a coin, rolling a four
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on a six-sided die, or running a mile in under six minutes.
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Take flipping a coin for example.
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There isn't only one single probability involved.
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Since there are two possible outcomes, getting heads
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or getting tails, that means we have two possible events
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and we need to assign probabilities to each one.
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When dealing with uncertain events, we are seldom satisfied
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by simply knowing whether an event is likely or unlikely.
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Ideally, we want to be able
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to measure and compare probabilities
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in order to know which event is relatively more likely.
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To do so, we express probabilities numerically.
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Even though we can express probabilities
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as percentages or fractions, conventionally, we write them
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out using real numbers between zero and one.
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So instead of using 20% or one fifth, we prefer 0.2.
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All right, now let us briefly talk
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about interpreting these probability values.
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Having a probability of one expresses absolute certainty
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of the event occurring and a probability of zero
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expresses absolute certainty of the event not occurring.
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You probably figured this out, but higher probability
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values indicate a higher likelihood.
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Okay, as you can imagine, most events we are interested
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in would've a probability other than zero and one.
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So values like 0.2, 0.5 and 0.66
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are what we generally expect to see.
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Even without knowing any of this, you can tell some events
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are more likely than others.
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For instance, your chance of winning
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the lottery isn't as great as winning a coin toss.
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That's why you can think of probability as a field
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that is about quantifying exactly
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how likely each of those events are on their own
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and that's what this course is going to teach you.
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So how about we start right away?
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Let's get into it.
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Generally, the probability of an event, a occurring denoted
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P Of A is equal to the number of preferred outcomes
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over the total number of possible outcomes.
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By preferred we mean outcomes that we want to see happen.
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A different term people use for such outcomes is favorable.
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Similarly, sample space is a term used
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to depict all possible outcomes, going forward
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we shall use the respective terms interchangeably.
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We will go through several examples
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to ensure you understand the notion well.
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Say event A is flipping a coin and getting heads.
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In this case, heads is our only preferred outcome.
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Assuming the coin doesn't just somehow stay
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in the air indefinitely
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there are only two possible outcomes, heads or tails.
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This means that our probability would be a half.
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So we write the following,
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p of getting heads equals one half, which equals 0.5.
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All right.
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Now imagine we have a standard six-sided die
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and we want to roll a four.
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Once again, we have a single preferred outcome
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but this time we have a greater number
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of total possible outcomes,
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six, therefore, the probability of this event
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would look as follows, P of rolling four equals one sixth
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or approximately 0.167.
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Great, events can be simple or a bit more complex.
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For example, what if we wanted to roll
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a number divisible by three?
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That means we need to get either a three or a six,
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so the number of preferred outcomes becomes two.
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However, the total number of possible outcomes stays
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the same since the die still has six sides.
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Therefore, we conclude that the probability
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of rolling a number divisible by three equals two
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over six which is approximately .33.
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So far, so good.
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Note that the probability of two independent events
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occurring at the same time is equal to the product
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of all the probabilities of the individual events.
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For instance, the likelihood
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of getting the ace of spades equals the probability
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of getting an ace times the probability of getting a spade.
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In a later lecture, we are going to define what we mean
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by independent, but for now, let's observe some
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more examples of probability.
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What about the probability of winning the US lottery?
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Even though it sounds
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like something that is completely different
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it actually follows the same idea.
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You take the number of preferred outcomes
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and divide it by all outcomes.
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Now, the number of preferred outcomes we have would be equal
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to the amount of different tickets we bought.
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The total number of possible outcomes
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on the other hand is just something we will learn
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how to calculate less than an hour from now.
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For the moment, just assume that there exists upward
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of 175 million outcomes for the US lottery.
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Therefore, each individual ticket only has a probability
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of winning equal to one
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over 175 million or approximately 0.000000005.
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How would your chances improve if you bought two tickets?
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How about five?
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I don't know about you
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but I like my odds of flipping a coin a lot more.
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Now that you know what probabilities are
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some of you might be wondering
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how and when we can use them, in the next video,
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we are gonna do that by introducing expected values.
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Thanks for watching.
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