exercises involving joint pmfs. Let \(X\) be the number of butterflies in the second sample that have tags (because they were also caught in the first sample). Probability Mass Functions (PMFs) In the following example, we compute the probability that a discrete random variable equals a specific value. belonging to the support of Notice that part of the specification of \(p(x)\) involves the possible values of \(x\); \(p(x)\) is only nonzero for \(x=2,3, \ldots, 8\). This distribution is known as Benfords law. We will study more properties and uses of Binomial distributions later. See below. Poisson distributions are often used to model random variables that count relatively rare events. There will be a whole number ( 0, 1, 2), numbers with variables ( 1y, 2y 3y) and numbers which are squared ( 2 y2, 3 y2 ). p_{X}(x) & = \binom{n}{x} p^x (1-p)^{n-x}, & x=0, 1, 2, \ldots, n
Example: If five coins were tossed and the outcome is taken independently then what would be the probability for number of heads occurred. 5 Facts You Should Know. ) ( p1**n1 p2**n2 . Solved Examples. A Swiss mathematician James Bernoulli define the Bernoulli random variable as a random variable having either success or failure as only two outcomes for the random experiment. I love to contribute to Lambdageeks to make Mathematics Simple , Interesting & Self Explanatory for beginners as well as experts. Think of a pmf as providing a compact formula for constructing the table/spinner. The consent submitted will only be used for data processing originating from this website. In order to get the entire marginal, we need to compute Thus, for example, PX(1) shows the probability that X = 1. The electric field is generated due to the charged particle. First, when the case is equal to zero. Having 12 years of experience in teaching. Figure 4.10: Probability mass functions for Binomial(10, \(p\)) distributions for \(p = 0.1, 0.3, 0.5, 0.7, 0.9\). But lets start with a simpler, but unrealistic, example where there are \(N=52\) butterflies, \(N_1 = 13\) are tagged and \(N_0=52-13 = 39\) are not, and \(n=5\) is the size of the second sample. thatwhere 0, & \text{otherwise}
The function \(\mu^x / x!\) defines the shape of the pmf. We could have written the pmf as, \[
We saw a similar idea in Example 1.5. for discrete random variable with probability mass function is the cumulative distribution function (c.d.f.) Now let's take a look at an example of a p.m.f. . Then calculate: P[Y 7]. This article will elucidate whether the electric field is a scalar or a vector quantity. Example I toss a fair coin twice, and let X be defined as the number of heads I observe. Compute the probability that the last butterfly selected is tagged but the others are not. \], \[
\end{align*}\]
The countable set of possible values of a discrete random variable \(X\), \(\{x: \textrm{P}(X=x)>0\}\), is called its support. notation: The joint pmf can be used to derive the Geometric distribution describes the number of trials needed to get one success. Find the range of X , RX, as well as its probability mass function PX. In PDF, the answer lies between variables that are in a continuous random order. Think of \(p(x)\) as a compact way of representing Table 4.1. Here If X is the random variable representing the defective products then it is of the binomial type with n=10 and p=0.01 then the probability that the pack will return is. \end{cases}
\left(\frac{13}{52}\right)\left(\frac{39}{52}\right)^4 = 0.079
Note that we only have whole numbers, i.e. Example: (chuck-a-luck/ wheel of fortune) In a specific game of fortune in hotel a player bet on any of the numbers from 1 to 6, three dice then rolled and if the number appears bet by the player once, twice or thrice the player that much units means if appear once then 1 unit if on two dice then 2 units and if on three dice then 3 units, check with the help of probability the game is fair for the player or not. Figure 4.2: The marginal distribution of \(Y\), the larger (or common value if a tie) of two rolls of a fair four-sided die. They are often, but not always, counting variables (e.g., \(X\) is the number of Heads in 10 coin flips). \end{align*}\], # plot the theoretical Binomial(5, 0.25) pmf, An Introduction to Probability and Simulation, The most home runs ever hit in a baseball game is 13, reliability of reported COVID-19 cases and deaths, Donnys solution is incomplete; he forgot to specify the possible values. 2. So the total integral over the probability function f (x) resolves to one. Here are the two dissimilarities between them: The PMF means the answer lies between variables that are in a discrete random order. The concept is very similar to mass density in physics: its unit is probability per unit length. PMF or probability mass function is a simple concept in mathematics. discrete random vector. S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} Given that X denotes the number of tails. Probability Mass Function is otherwise referred to as Probability Function or frequency function. If a and b are constants then E [aX + b]=aE [X] + b This we can understand easily as =aE [X] + b }, \;\qquad x=0,1,2,\ldots\\
Therefore the probability of rolling a 3 is f(3) = 1/6. The probability also needs to be non-negative. values it can take) the joint pmf can also be written using the following alternative . \text{SD of $X$} & = \sqrt{\mu}
\forall x \in X, p (x) \geq 0 x X,p . e^{-2.3} \frac{2.3^x}{x! Given below are the steps that you need to follow to find the PMF of a variable: Start solving the question by fulfilling the first condition of the PMF. The sum of the probabilities is equal to unity (1). p_X(x) =
Randomly select \(n\) tickets from the box with replacement and let \(X\) be the number of tickets in the sample that are labeled 1. There is not one Poisson distribution but rather a family of Poisson distributions. The probability of a head is denoted as "p" whereas "k" represents the count of the coin tosses till the head is obtained. \], \(\textrm{P}(X=3)=p_X(3)=e^{-2.3}2.3^3/3! To simplify the display we have lumped 4 and 5 into one 4+ category. Examples of Mass Functions and Densities For a given state space, S, we will describe several of the most frequently encountered parameterized families of both . Once we have the shape of the distribution, we can renormalize by multiplying all values by a constant, in this case \(e^{-2.3}\), so that the values sum to 1. Figure 4.5: Spinner corresponding to the Poisson(2.3) distribution. Its given because when you are drawing the variable, it produces distinct outcomes or results. Since the selections are independent, Since the selections are independent, the probability of the outcome SFFFF is, The probability of the outcome FFFFS is the same as in the previous part, Each of the particular outcomes with 1 tagged butterfly has probability, Each of the particular outcomes with 2 tagged butterflies (like SSFFF) has probability, Similar to the previous part. Figure 4.5 displays a spinner corresponding to the Poisson(2.3) distribution. In English language, the verb may falls under the modal auxiliary verbs. Probability Mass FunctionAdvance Computer Architecture (ACA): https://www.youtube.com/playlist?list=PLz8TdOA7NTzSOHaom_1AGQNrVeAzI3GIMAnalysis and design of algorithms (ADA) (DAA): https://www.youtube.com/playlist?list=PLz8TdOA7NTzQkI_TCASvO8Abbo9XgUuiuComputer organization and architecture - COA: https://www.youtube.com/playlist?list=PLz8TdOA7NTzQNlzLxRfsv2KexBzRSn3MFTheory of computation (TOC): https://www.youtube.com/playlist?list=PLz8TdOA7NTzTdFM93ak-qBVpX1aMFuw9cSoftware Engineering -SE: https://www.youtube.com/playlist?list=PLz8TdOA7NTzQJ62qHgIfwZmRWt-EAMn-WDBMS/tutorials for beginners- Database Management System: https://www.youtube.com/playlist?list=PLz8TdOA7NTzToibk7WXtD5qvBMnkZkwh7Operating system Tutorials -OS: https://www.youtube.com/playlist?list=PLz8TdOA7NTzSwXreT70Z3CXfD6KCVugUnOops- Object oriented programming concepts: https://www.youtube.com/playlist?list=PLz8TdOA7NTzSe5HdXz_nAk9kmOJLDi5ZBMathematics-III/BTech/IVsem/RGPV syllabus: https://www.youtube.com/playlist?list=PLz8TdOA7NTzSQl-MA9tyiJqHSX1wpHPyKData structures- DSA: https://www.youtube.com/playlist?list=PLz8TdOA7NTzSMvlDZc4F8r71o4FKaRPU5Compiler Design lectures: https://www.youtube.com/playlist?list=PLz8TdOA7NTzSu-ePbGtghroiomWA-FFK9Discrete Mathematics/ Discrete Structures: https://www.youtube.com/playlist?list=PLz8TdOA7NTzTk-kaYf8NVfMM4ZDNQKDgAInternet and Webtechnology https://www.youtube.com/playlist?list=PLz8TdOA7NTzRAOymI4mSobv6Q0cS91og- \[\begin{align*}
Example: The probability mass function for X Watch on This example illustrated the tabular and graphical forms of a p.m.f. characterizes the distribution of a and mathematical expectation for such random variable we defined was, we now see some of the results of mathematical expectations, Example: for the following probability mass functions find the E(X3), E (X3) = (-1)3
0.2 + (0)3 0.5 + (1)3 * 0.3, In the similar way for any nth order we can write, For the mean denoted by the variance of the discrete random variable X denoted by var(X) or in terms of expectation will be. That is, the expectation of a Bernoulli random variable is the probability that the random variable equals 1. is: If the three values have the same probability, then the joint probability mass pk**nk ) P = ( 4! Instead, we can usually define the probability density function (PDF). This is a straightforward multi-variate generalization of the definition of the probability mass function of a discrete variable (uni-variate case). Probability Mass Function integrates that any given variable has the probability that the random number will be equal to that variable. P = Probability of the random variable when it equals xk. Plug each possible value into the pmf from the previous part. Let us here find out how the verb may, can be changed in to the passive voice. (1/6)=(7/2), E[X2] =12.(1/6)+22.(1/6)+32.(1/6)+42.(1/6)+52.(1/6)+62. = e^{-2.3}e^{2.3} = 1
\int f (x)dx = 1 f (x)dx = 1. We and our partners use data for Personalised ads and content, ad and content measurement, audience insights and product development. As this type of random variable is obviously discrete so this is one of discrete random variable. So the probability mass function for the Bernoulli random variable is. Suppose you want to estimate \(N\), the number of monarch butterflies in Pismo Beach. We would expect the long run average value to be. Suppose Use the fact that a negative binomial random variable Negbin(r,p . f (x) = { k(7x+ 3) 0 if x = 1,2 or 3 otherwise Each of the computations involves a sum over all the possible values of X can take values 0 (No tail) or 1 (One tail) or 2 (two tails) or 3 (three . Probabilities for other general events, e.g., \(\textrm{P}(X \le x)\) can be obtained by summing the pmf over the range of values of interest. separately for each in the similar way we can calculate for any such experiment. The joint pmf can also be used to derive the conditional probability mass where p is the probability of success and 1-p will be the probability of failure. The two important concept related to the discrete random variables are the probability of discrete random variable and distribution function we restrict the name to such probability and distribution function as. (1/6) =(1/6)(91), One of the important identity for variance is. p_X(x) = \binom{5}{x}\left(\frac{13}{52}\right)^x\left(\frac{39}{52}\right)^{5-x}, \qquad x = 0, 1, 2, 3, 4, 5
3. As already we now familiar with the discrete random variable, it is the random variable which takes countable number of possible values in a sequence. We have to find P (2 < X < 3). If \(X\) has a Binomial(\(n\), \(p\)) distribution then
P (a<x<b) = ba f (x)dx = (1/2)e[- (x - )/2]dx. p_Y(y) =
The pmf of \(Y\) is, \[
We see from this that the sample mean is what maximizes the likelihood function. We can see that the Poisson(2.3) distribution models the data reasonably well. \text{SD of $X$} & = \sqrt{np(1-p)}
For example, tossing a coin until the 1st head turns up. An example of the binomial distribution is the probability of getting exactly one 6 when someone rolls a fair die three times. The CDF of a random variable X X is a function that represents the probability that X X will be less than or equal to x x. discrete random vector. It is a part of statistics. If not, why not? The unconditional/marginal probability of success is the same on every trial, and equal to. \text{SD of $X$} & = \sqrt{\mu}
Where. Random vectors, where Thus, pred = 0.2, pgreen = 0.3, and pblue = 0.5 We plug these inputs into the multinomial formula, as shown below: P = ( n! \text{Long run average value of $X$} & = \mu\\
Home Maths Notes PPT [Maths Class Notes] on Probability Mass Function Pdf for Exam. In practice, \(N\) is unknown and the point of capture-recapture sampling is to estimate \(N\). Solved Example 1: Let X be a random variable, and P (X=x) is the PMF given below; 1. Certain common distributions have special names. When (1) Find the expectation of the Bernoulli random variable X with probability p. (2) Find the variance of X. Solved Problems 14.1 Probability review Problem 14.1. With 1/0 representing S/F, we can also obtain the number of successes with X = RV(P, sum). Step 4: As you start using simultaneous equations, you will get two answers in the end. A fair die is tossed 7 times. pX (k) = (1 p)k1p. Online appendix. Then \(X\) has a Binomial(\(n\), \(p\)) distribution. Find the pmf probability of (i) P (X 6 ) Solution: Given: P (xi)=1 1. In particular, in R the d commands (dbinom, dnorm, etc) are used for both discrete and continuous distributions. Definition 4.2 A discrete random variable \(X\) has a Poisson distribution with parameter103 \(\mu>0\) if its probability mass function \(p_X\) satisfies
We have seen that a distribution of a discrete random variable can be represented in a table, with a corresponding spinner. (adsbygoogle = window.adsbygoogle || []).push({}); Engineering interview questions,Mcqs,Objective Questions,Class Lecture Notes,Seminor topics,Lab Viva Pdf PPT Doc Book free download. Here are two conditions on which the probability function should fall upon: The definition of Probability Mass Function is that its all the values of R, where it takes into argument any real number. Solution: We know that success probability P (X = 1) = p = 0.6. Many people use PMF to calculate two main concepts in statistics- mean and discrete distribution. are denoted by https://www.statlect.com/glossary/joint-probability-mass-function. \]. The term joint probability function is often used as a The conditional probability Pr { A | B } of the event A given the event B is defined by. This is shown by the Fundamental Theorem of Calculus. In particular, if the leading digits in a series of values follows a distribution other than Benfords law, such as discrete uniform, then there is evidence that the values might have been fudged. You can evaluate the pmf at multiple values. The PDF is the density of probability rather than the probability mass. Having the immense ability of problem designing and solving. "Joint probability mass function", Lectures on probability theory and mathematical statistics. \], \[\begin{align*}
The variables are in random continuous order. Step 2: Take all the values of P ( X- x) and add it up. Here are some of the places where theres an application of PMF: One of the sections where PMF is used is statistics. Definition Example 4.6 Randomly select a county in the U.S. Let \(X\) be the leading digit in the countys population. 5 Facts You Should Know, probability mass function for discrete random variables, mathematical expectation for such random variable, For the arbitrary constants a and b we have. Example 4.4 Let \(Y\) be the larger of two rolls of a fair four-sided die. We use pmf for discrete distributions and reserve pdf for continuous probability density functions. For a thorough discussion of joint pmfs, go to the lecture entitled p_X(x) =
\frac{4-|x-5|}{16}, & x = 2, 3, 4, 5, 6,7, 8,\\
All the probabilities for the given discrete random variables provided by Probability Mass Function. For example, if the countys population is 10,040,000 (Los Angeles County) then \(X=1\); if 3,170,000 (Orange County) then \(X=3\); if 283,000 (SLO County) then \(X=2\); if 30,600 (Lassen County) then \(X=3\). Integrating x + 3 within the limits 2 and 3 gives the answer 5.5. Solution: here we know when die thrown the probabilities for each face will be, hence for calculating variance we will findexpectation of random variable and its square as, E[X]=1.(1/6)+2.(1/6)+3.(1/6)+4.(1/6)+5.(1/6)+6. The probability mass function example is given below : Question : Let X be a random variable, and P (X=x) is the PMF given by, Determine the value of k Find the probability (i) P (X 6), (ii) P (3<x 6 ) Solution : (1) We know that; P (x i )=1 Therefore, 0 + k + 2k + 2k + 3k + k 2 + 2k 2 + 7k 2 + k = 1 9k + 10k 2 = 1 10k 2 + 9k - 1 = 0 (i.e., over the support The shape of the graph of a probability density function is a bell curve. The function \(p(x)\) is called the probability mass function of the discrete random variable \(X\). \sum_{x=0}^\infty e^{-2.3} \frac{2.3^x}{x!} Why might we assume this particular Poisson(2.3) distribution for the number of home runs per game? Benfords law has been used recently to test reliability of reported COVID-19 cases and deaths. The parameter 5 is the size of the sample, and the parameter 0.25 is the proportion of successes in the population. The Binomial random variable is having the name Binomial because it represent the expansion of. p_X(x) =
We and our partners use cookies to Store and/or access information on a device. Solution Exercise 2 For example, suppose we roll a dice one time. Careful: Dont confuse the number \(p\), the probability of success on any single trial, with the probability mass function \(p_X(\cdot)\) which takes as an input a number \(x\) and returns as an output the probability of \(x\) successes in \(n\) Bernoulli(\(p\)) trials, \(p_X(x)=\textrm{P}(X=x)\). = 0.203\), \(\textrm{P}(X = 5) =p_X(5)=e^{-2.3}2.3^5/5! Figure 4.8: Spinner corresponding to the Binomial(5, 0.25) distribution. The second time is when the value is negative, the value of the probability function is always positive. \begin{cases}
; each column corresponds to one of the possible values of Figure 4.1: The marginal distribution of \(X\), the sum of two rolls of a fair four-sided die. , 2. Benfords law is often used in fraud detection. Probability Function shows the various probabilities of the discrete variable data. The is proportional to specification defines the shape of the plot; the constant just rescales the values on the probability axis. You should simplify your answer to a fraction that does not involve binomial coecients. \end{align*}\], \[\begin{align*}
The word mass is used to denote the expectations of discrete events. The terms pdf and density are sometimes used in both discrete and continuous situations even though the objects the terms represent differ between the two situations (probability versus density). It plays a vital and essential role in the study of statistics. \end{cases}
The conditional probability mass function pX|Y ( x | y) of X given Y = y is . Find the probability mass function of Y. Example of a discrete random variable: Let Y be the random variable of a function, and this is its probability mass function: Py (y) = P (Y-y), for all y belongs to the range of Y. The connecting dots displayed by Poisson(2.3).plot() are determined by the theoretical Poisson(2.3) pmf. \begin{cases}
For example, let \(X\) be the sum of two rolls of a fair four-sided die. For now well just present Figure 4.6 which displays the actual distribution of home runs over the 2431 games in the 2018 MLB season. Example 2: If a Bernoulli distribution has a parameter 0.45 then find its mean. Compute the probability that the first butterfly selected is tagged but the others are not. \frac{\textrm{P}(X=3)}{\textrm{P}(X=5)} = \frac{p_X(3)}{p_X(5)} = \frac{e^{-2.3}2.3^3/3!}{e^{-2.3}2.3^5/5!} mass functions of the single entries of the random vector. At a later date, you then capture a second sample of \(n\) butterflies, selected randomly104 with replacement. which is the posterior probability. This is a straightforward multi-variate generalization of the definition of (Assume that \(N\) is a fixed but unknown number; the population size doesnt change over time.) \[\begin{align*}
In a certain company the probability of defective is 0.01 from the production. 3. Figure 4.4: Impulse plot representing the Poisson(2.3) probability mass function. Specifically, for X=a, P(a)=P(X=a) is its p.m.f. Let S (for success) represent that a selected butterfly is tagged, and F (for failure) represent that its not tagged. All the probability characteristics for the probability will obviously applicable to probability mass function like positivity and summation of all p.m.f will be one e.t.c. Derive the moment generating function of , if it exists. Since the tickets are labeled 1 and 0, the random variable \(X\) which counts the number of successes is equal to the sum of the 1/0 values on the tickets. An example of data being processed may be a unique identifier stored in a cookie. ). There will be a whole number ( 0, 1, 2), numbers with variables ( 1y, 2y 3y) and numbers which are squared ( 2 y. \end{align*}\], \(p_Y(x) = \frac{2x-1}{16},\; x = 1, 2, 3, 4\), \[
\text{Long run average value of $X$} & = np\\
pmfs. However, we use \(\mu\) to denote the parameter of a Poisson distribution, and reserve \(\lambda\) to denote the rate parameter of a Poisson process (which has mean \(\lambda t\) at time \(t\))., Well compare to the case of sampling without replacement later., \[\begin{align*}
The function is defined as F X(x) = P (X x) F X ( x) = P ( X x). Example 3.2.1 Continuing in the context of Example 3.1.1, we compute the probability that the random variable X equals 1. marginal probability \], \(\textrm{P}(X \ge 3) = 1 - \textrm{P}(X <3) = 1 - (0.301 + 0.176) = 0.523\), \[
is the probability \frac{2.3^x}{x! Discrete random variables take at most countably many possible values (e.g. \end{align*}\], # plot the simulated values and their relative frequencies, \[
discrete random vector and that its Figure 4.7: Impulse plot representing the Binomial(5, 0.25) probability mass function. Engineering 2022 , FAQs Interview Questions. \]. For each possible value \(x\) of the random variable \(X\), \(\textrm{P}(X=x)\) can be obtained from the following formula, \[
The probability distribution function is essential to the probability density function. That is, for each selection the conditional probability that a butterfly is tagged is 13/52 regardless of the results of other selections. Below we use the Python package tabulate to construct a somewhat nicer table. "Probability Mass Function" is calculated for discrete random variable. I.e. = \frac{2.3^3/3!}{2.3^5/5!} The Probability Mass Function (PMF) provides the probability distribution for discrete variables. \[\begin{align*}
The parameter to fit our model should simply be the mean of all of our observations. . The axioms of probability imply that a valid pmf must satisfy \[\begin{align*}
This function is extremely helpful because it apprises us of the probability of an affair that will appear in a given intermission. The general formula for probability mass function is as follows: - PX(xk) = P (X = xk) for k = 1,2,k where, X = Discrete random variable. be a For example, \(\textrm{P}(X = 2) = 1/16 = p(2)\); \(\textrm{P}(X=5)=4/16=p(5)\); \(\textrm{P}(X=7.5)=0=p(7.5)\). Your email address will not be published. Let X be a random variable, and P (X=x) is the PMF given below; K= -1 is not the desired answer because the pmf probability value lies between 0 and 1 ( its one of the conditions which is mentioned above), ---- >> Below are the Related Posts of Above Questions :::------>>[MOST IMPORTANT]<, Your email address will not be published. Dont get this tabulate confused with .tabulate() in Symbulate. \[
Discrete random variables take at most countably many possible values (e.g. The approximate long run average value and variance are both about equal to the parameter 2.3. is. Its expected value is. \], \(\left(\frac{13}{52}\right)\left(\frac{39}{52}\right)^4=0.079\), \[
\begin{cases}
Figure 4.9: Probability mass functions for Binomial(\(n\), 0.4) distributions for \(n = 5, 10, 15, 20\). Compare the approximate \(\textrm{P}(X = 2)\) to the theoretical value. Let us discuss its formula: . & = \frac{e^{-\mu}\mu^x}{x! \textrm{P}(X = 1) = \binom{5}{1}\left(\frac{13}{52}\right)\left(\frac{39}{52}\right)^4 = 5(0.079) = 0.3955
The simple meaning of Probability Mass Function is the function relating to the probability of those events taking place or occurring. \end{align*}\]. the random vector will be equal to that point. That's it. Example 4.5 Donny Dont provides two answers to Example 4.4. Example: when a die is thrown calculate the variance. If we assume there will be no unfair means with the dice and con techniques then by assuming the outcome of the dice independently the probability of success for each dice is 1/6 and failure will be, 1-1/6 so this turns to be the example of binomial random variable with n=3, so first we will calculate the winning probabilities by assigning x as players win, Now to calculate the game is fair for the player or not we will calculate the expectation of the random variable. Construct a table, plot, and spinner representing the distribution of, Make an educated guess for the long run average value of. [Maths Class Notes] on Cumulative Distribution Function Pdf for Exam, [Maths Class Notes] on Probability and Statistics Pdf for Exam, [Maths Class Notes] on Probability Distribution Pdf for Exam, 250+ TOP MCQs on Random Variables and its Probability Distributions | Class 12 Maths, [Maths Class Notes] on Random Variables Pdf for Exam, [Maths Class Notes] on Mean and Variance of Random Variable Pdf for Exam, 250+ TOP MCQs on Probability Distributions and Answers, [Maths Class Notes] on Multiplication Rule Probability Pdf for Exam, [Maths Class Notes] on Conditional Probability Pdf for Exam, [Maths Class Notes] on Normal Distribution Pdf for Exam, [Maths Class Notes] on Uniform Distribution Pdf for Exam, [Maths Class Notes] on Types of Events in Probability Pdf for Exam, [Maths Class Notes] on Probability Questions Pdf for Exam, [Maths Class Notes] on Probability For Class 10 Pdf for Exam, [Maths Class Notes] on Probability Pdf for Exam, [Maths Class Notes] on Chance and Probability Pdf for Exam, [Maths Class Notes] on Coin Toss Probability Formula Pdf for Exam, [Maths Class Notes] on Multiplication Theorem of Probability Pdf for Exam, [Maths Class Notes] on Probability Worksheets Pdf for Exam.