Conditional probability Definition and 242 Threads

Product Testing A supposed coffee connoisseur claims she can distinguish between a cup of instant coffee and a cup of drip coffee 75% of the time. You give her 5 cups of coffee and tell her that you will grant her claim if she correctly identifies at least 4 of the 5 cups. (a) What are her...

Homework Statement I have two kids. Given that at least one of them is a girl, what is the probability that they are both girls? Homework Equations The Attempt at a Solution I think it's 1/3, because the possibilities are: Boy Boy, Boy Girl, Girl Boy, Girl Girl. There are three...

Hello, I'm somewhat confused by the expression f(X = x | Y = y) = \frac{f(X=x)}{f(Y=y)} (which, if I'm right, is the definition of a conditional probability density? My course seems to state it as a theorem, without proof, but then again my course is a little bit vague; although I welcome...

Homework Statement Basically, I'm given the probability of 4 independent events: P(A) = 0.04 P(B) = 0.03 P(C) = 0.02 P(D) = 0.01 If anyone of these occur, a failure will happen. More than one can happen at the same time. I need to find the probability that more than one of them...

Homework Statement A bowl contains 10 chips: 6 red chips and 4 blue chips. three chips are drawn at random and without replacement. Compute the conditional probability that a) 2 are red and one is blue; given that at least 1 red chip is among the 3 selected b) all are red, given that at least...

Homework Statement You are a member of a class of 18 students. A bowl contains 18 chips: 1 blue and 17 red. Each student is to take 1 chip from the bowl without replacement. The student who draws the blue chip is guaranteed an A for the course. a)If you have the choice of drawing first...

Homework Statement An urn contains four colored balls: two orange and two blue. Two balls are selected at random without replacemen, and you are told that at least one of them is orange. What is the probability that the other ball it orange? Homework Equations Now, we have been...

Homework Statement romeo proposed to juliet. now he's waiting for her response. R = 'event that she replies' W='event that she wants to get married' Mon = 'event on monday' Tue = 'event on Tuesday' P(R\wedgeMon | W) = 0.2 P(R\wedgeTue | W) = 0.25 P(R\wedgeMon| \bar{W}) = 0.05...

I'm wondering how conditional probability relates to concepts of sample space, observation space, random variable, etc. Using the notation introduced in the OP here, how would one define the standard notation for conditional probability "P(B|A)" where A and B are both subsets of some sample...

The ABC manufacturing company purchases a certain part from two suppliers (A supplier & B supplier). The parts received from the suppliers are used to produce the company’s major product. The parts received from the suppliers are either defective or good parts. The A supplier provides 80% of all...

Hi friends, The problem: Assume the events A_i for i=1,2,...,n are of the same type. please trace the following relations. P(X│A_1 A_2…A_n )= \frac{P(X)}{P(A_1 A_2…A_n )} P(A_1 A_2…A_n│X)= \frac{P(X)}{P(A_1 A_2…A_n )}[ (-1)^{n-1} P(⋃_{i=1}^{n} A_i │X)+(-1)^{n-2} {n \choose 1}...

Hi everyone, The problem: Is this relation true? If so, how (or maybe where) it could be proved?P(A│B∪C)≤P(A│B)+P(A│C)-P(A|BC) and what about its possible generalization? thanks a lot in advance.

The discrete random variable X has probability density P(X=x) =kp^x for x=0,1,... where p \in (0,1). Find normalizing constant k and E(X) as functions of p. For each integer x>0 find P(X>=x) and hence find P(X=y|X>=x) for each integer y>0. found k=1-p E(X)=\sum kxp^x =p/(1-p) P(X>=x) =...

Homework Statement A car dealer estimates that 50% of customers entering the dealership will buy a normal car, 20% will buy a high-end car, and 30% are just browsing. If 5 customers enter his dealership on a particular day, what is the probability that two will purchase high-end models, one...

How can I find the conditional probability density function of a dependent variable given the independent variable set. Say, Y is some deterministic function of a set of variables X , or Y=f(X)+e. How can I fine the conditional pdf or P(Y|X) ?

How would I evaluate the following: P(A|(B or C)) where B and C are two mutually exclusive events. I have scoured a couple texts and the internet but have made no headway, any insight would be greatly appreciated. QN

Consider this scenario: "From families with three children, a family is selected at random and found to have a boy. What is the probability that the boy has an older brother and a younger sister? Assume that in a three-child family all gender distributions have equal probabilities."...

Consider the conditional probability P(A|C) . Is it true that we can calculate above as Summation over B P(A|B)*P(B|C) .

I could have posted this in the homework section but I didn't see a section on probability. Also, this is more of a general conditional probability question asked through an example. A band called Radiohead is inspired by an old band called The Beatles. 50% of music critics think the beatles was...

A band called Radiohead is inspired by an old band called The Beatles. 50% of music critics think the beatles was a great (G) band, 40% that it was moderate (M) and 10% that it was awful (A). These critics have also compiled the following table:0.8 0.1 0.1 0.1 0.9 0 0.2 0.3 0.5The table says...

Homework Statement 99% of all babies survive delivery. However, 10 % of all births involve Cesarean (C) sections, and when a C section is performed the baby survives 98% of the time. If a randomly chosen pregnant woman does not have a C section, what is the probability that her baby survives...

Homework Statement The old TV game Let’s Make a Deal hosted by Monty Hall could be summarized as follows. Suppose you are on a game show, and you are given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows...

Homework Statement 4 balls are placed successively into 4 cells, all 4^4 arrangements being equally probable. Given that the first 2 balls are in different cells (event B), what is the probability that one cell contains exactly 3 balls (event A)? Homework Equations P(A|B) = P(A...

So i a need to find E[XY], expected value of XY But the process of finding E[X] includes a long and tideous integral which i am trying to avoid. So computing E[XY] using its formula is also something i am trying to avoid. But could i use this identity? E[XY] = E[E(XY/Y)] = E[Y[E(X/Y)]]...

Given the following: P(shop) = 0.25 P(rob) = 0.02 P(enter-store | !shop, !rob) = 0.01 P(enter-store | shop, !rob) = 0.80 P(enter-store | !shop, rob) = 0.50 P(enter-store | shop, rob) = 0.90 P(point-gun | rob) = 0.70 P(point-gun | !rob) = 0.01 Find: a) P(enter-store) b) P(shop | enter-store) c)...

Homework Statement Students who actively participate in class are 4 times more likely to get a good grade than those who don't. 15% of students actively participate in class; 20% of students get a good grade. A person did not participate in class, what is the probability that he got a good...

Homework Statement A radioactive source emits particles according to a Poisson process, at an average rate of λ per unit time. Each particle emitted has probability p of being detected by an instrument, independently of other particles. Let X be the number of particlese emitted in a given...

Hi everybody. I keep on reading Rohatgi's book "An introduction to Probability and Statistics" and I have worked out the following problem: "An urn contains r red marbles and g green marbles. A marble is drawn at random and its color noted. Then the marble drawn, together with c > 0 marbles of...

Homework Statement In a neighbourhood, 75% of households speak only English at home, and 5% speak only Spanish at home. Four households are randomly chosen. Given that at least one of the four households speaks only Spanish at home, what is the probability that only one speaks Spanish at...

Dear all, I am in need of an advice regarding a conditional probability... Although it is very simple, I haven't found anywhere anything similar so I am not sure I get it right...I tried numerically calculating this conditional probability using 2 approaches (in Mathematica) and I get 2...

Hi I'm having trouble doing this question, any help would be much appreciated The time needed to defuse the bomb is modeled by a continuous uniform random variable A in the interval (0,2) The the lifetime of the bomb is modeled by a random variable B. Given A=a its assumed that B has a...

Homework Statement 1 and 2 take turns shooting at a target. Each time 1 shoots he hits the target with probability p1; 2 hits it with probability p2 each time. 1 shoots first. They stop when the target has been hit twice. What is the probability that... a) The first hit was by 1 ? b) Both...

Homework Statement The masses of suspended particles in a sample of water taken from a lake can be assumed to be a random variable which is normally distributed with mean 2.17 and variance 0.979. Find the probability that out of 4 samples of lake water known to contain less than 1.8 mg of...

Homework Statement A truth serum has the property that 90% of the guilty suspects are properly judged while 10% are improperly found innocent. On the other hand innocent suspects are misjudged 1% of the time. If the suspect is selected from a pool of suspects of which only 5% have ever...

Homework Statement I'm currently trying to revise for exams and really struggling on this problem: Suppose you have 3 coins that look identical (ie don't know which is which) with probabilites of 1/4, 1/2 and 3/4 of showing a head. 1. If you pick a coin at random and flip it, what is...

Homework Statement X and Y two independent random variables with distribution U(0, 1/2). Find the density of (X + Y)2|X - Y > 0 The Attempt at a Solution I was hoping this would be simpler, but somehow I always end up with nothing. The only thing I can work out just fine is that P(X...

Homework Statement So I just took a probability test and I'm having a hard time with the fact that my answer is wrong. I've done some research online and I believe I am correct, I was hoping to get some input. I'm new to using LaTeX so sorry if it's sloppy. Thanks! Problem: Suppose that the...

Hallow! May anyone please help me on conditional probability i actualy understand how it occurs but the formula? The logic behind it! ie P(A/B)=P(AnB)/P(B)

Homework Statement For the random variable X with the following cumulative distribution function: Calculate P(X\leq1.5|X<2), P(X\leq1.5|X\leq2) and P(X = -2| |X|=2) The Attempt at a Solution This is an exercise about a subject I'm yet to see in class, but the teacher asked us to...

Homework Statement One satellite is scheduled to be launched form Cape Canaveral in Florida, and another launching is scheduled for Vandenberg AFB in California. Let A denote the event that the Vandenberg launch goes off on schedule, and let B represent the event that the Cape Canaveral...

Homework Statement Components of a certain type are shipped to a supplier in batches of ten. suppose that 50% of all such batches contain no defective components, 30% contain one defective component, and 20% contain two defective components. Two components from a batch are randomly selected...

Hi all, I've got this very simple problem: I know it is an elementary problem, but I never really got into that bayes' theorem, which I need to use here, right? I would be grateful for simple and plain explanation. thanks for your time, rahl.

box A contains 2 red balls, box B contains 2 white balls, box C contains 1 red ball and 1 white ball, A box is selected at random (with equal probabilities) and one ball is taken at random from that box. (i) Compute the probability of selecting a white ball using Bayes Rule. A(1/3)...

The problem is stated as follows: You have an urn with 12 red balls, 20 green balls, and 13 yellow balls. Suppose 3 balls are drawn without replacement. What is the probability the third ball is yellow given the first ball is green. I am pretty sure I use Bayes formula, but I am not certain...

Homework Statement Chose a point at random in a square with sides 0<x<1 and 0<y<1. Let X be the x coordinate and Y be the y coordinate of the point chosen. Find the conditional probability P(y<1/2 / y>x). Homework Equations No clue. The Attempt at a Solution Apparently, according...

An event A has a 70% chance of occurring, an event B has a 40% of occurring and the event (A n B) has a 20% chance of occurring. The events are not unrelated. If A does not occur what is the probability of B? notation A' means (not A) I have tried to use the formula P(B|A') = P(B n...

Homework Statement A baseball player compiles the following information : He hits a home run in 34% of his games He gets a strike out in 40% of his games In 78% of his games he hits a home run or his team wins In 10% of his games he hits a home run and gets a strike out In 26% of his...

Homework Statement There are w white balls and b black balls in a bowl. Randomly select a ball from the bowl and then return it to the bowl along with n additional balls of the same color. Another single ball is randomly selected from the bowl(now containing w+b+n balls) and it is black. Show...

Homework Statement A concrete beam my fail by shear or flexure. The failure probability in shear is equal to the failure probability in flexure, and the probability of failure in shear when the beam is loaded beyond is flexure capacity (ie, it has already failed in flexure) is 80%. Find the...

Which probability would you expect to be greater? (a) The probability that a person is smart given that he has a university degree, OR, (b) the probability that a person has a university degree given that he is smart? I would say (a)>(b), but, what would you say?