Lecture Note
University
Duke UniversityCourse
Data Science Math SkillsPages
3
Academic year
2023
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35
Joint Probabilities Joint probabilities are probabilities involving multiple events. For example, if you want toknow the probability of an event A and another event B occurring simultaneously, youwould use joint probabilities.A Venn diagram is a diagram that shows all possible logical relations between a finitecollection of different sets.The names of the sets are often displayed on the diagram, and where two or more setsoverlap, they are said to be members of common subset. Next, we will discuss joint probabilities. A joint probability is the probability that two eventsfrom two separate probability distributions are both true. So, our notation allows us to abbreviate this complex probability by combining the twocapital letters and separating them with a comma. This is read as "the joint probability of Aand B", which means the probability of both A and B being true. Note that the ordering of probabilities does not matter in joint probabilities. X = { x1, x2,x3, ..., xn} Y = {y1, y2,y3, ..., yn} Similarly, we use capital letters to refer to the entire probability distribution and lowercaseletters to refer to individual probabilities. We can reference a specific probability in thesame way. So, we can conclude that the joint probabilities are equal for both and regardless of whichorder we write the terms. Or, in other words, the same is true of each individual probabilitywithin the distribution. P(X, Y ) = P(Y, X) P(x1,y1) = P(y1, x1) Thus, the probability of X1 and Y1 occurring together is equal to that of Y1 and X1occurring together. This principle may seem obvious to you but it turns out to be a veryuseful one in many situations. In this section, we'll discuss the independence of two probability distributions. If twoprobability distributions are independent, then knowing the outcome of one does not alterour belief in the truth of the other. Thus, knowing one outcome does not alter theprobability of the other. When two probabilities are independent, they can be thought of asbeing unrelated and unconnected. Suppose I toss a coin and it has a 50 percent chance of coming up heads. And suppose Iroll a die and it has a 1 in 6 chance of coming up to show a 3, okay? The joint probability of three heads in six tosses of a fair coin and the number 3 showingup on a six-sided die is equal to the probability of getting three heads in six tosses, timesthe probability for rolling a three on a six-sided die. This is 1/2 times 1/6, or 1/12.
So, for probability problems involving independent distributions, this is the formula tocalculate the probability that both events happen at the same time. We refer to theprobability of two independent events occurring simultaneously as a product distribution. So, when the joint probability equals the product of the two separate probabilities, the twodistributions are said to be independent. We often use Venn diagrams to describe thissituation. These diagrams show the intersection and union of sets to illustrate the intersection of twosets of events. And, develop some intuition about what it means to say the probability thatboth events occur. Px1, y1= Px1Py1 For example, if we have the probability of flipping a coin and it comes up heads, and thenwe have the probability of tossing a die and it comes up three, we can represent this areaas the product of these two probabilities.
The universe of all possible outcomes is the set of all possible combinations of the relevantevents we have defined. The probability that both events occur at the same time iscalculated by multiplying their individual probabilities together. The Venn diagram may be more useful in explaining or demonstrating the concept ofprobability. Px1 or y1= Px1+Py1-P(x1, y1) What is the probability that we will get heads or tails? This can be written in two ways: witha "+" (meaning either one event or the other) or a "-" (meaning both events occur). So, we are interested in the probability that the coin lands on heads or the die lands onthree. If we simply add the two circles together, we will count the central area twice. To avoid thisdouble counting, we subtract the area of the circle from the sum of the two circles. We are subtracting the joint probability, which is equal to the product of the individualprobabilities 1/12. So, the probability that either the coin comes up heads or the dyecomes up 3 would be 1/2 + 1/6- 1/12 or 6/12 + 2/12- 1/12 = 7/12. Or
Joint Probabilities in Probability Theory: Coin and Die Experiment
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