Conditional probability Definition and 231 Threads

Homework Statement If you already have waited five minutes at the bus stop, what is the probability of having to wait additionally five minutes or more? Homework Equations Declare a random time T, together with the CDF shown below, which specify the time (in minutes) that a frozen traveller...

I'm having trouble understanding an example supposed to motivate Bayes' theorem. Assume that 40% of all interstate highway accidents involve excessive speed on part of at least one of the drivers (event E) and that 30% involve alcohol use by at least one drives (event A). If alcohol is involved...

So I was checking the How to self-study math thread and saw that someone suggested that It would be helpfull to create this kind of thread. And because we are writting a test on thursday on Probability I though it would be nice to find out which parts I still need to double-check. So these...

Homework Statement Suppose that 70% of the statisticians are shy, whereas 30% of the economist are shy. Suppose also that 80% of the people at a large gathering are economists and the remaining 20% are statisticians. If you randomly meet a person at the gathering and the person is shy, what is...

Homework Statement Urn I contains 3 white and 5 red balls, whereas urn II contains 2 white and 1 red ball. A ball is randomly chosen from urn I and put into urn II, and a ball is then randomly selected from urn II. (a) What is the probability that the ball selected from urn II is white? (b)...

Homework Statement I don't undersand why Pr(A l B) = P(AnB)/P(B) Homework Equations conditional probability[/B] Pr(A l B) = P(AnB)/P(B) A given B The Attempt at a Solution I can make a bunch of problems with this definition intuitively, but I really want to understand with sets what is...

Homework Statement Consider a bag containing 10 balls of which a few are black balls. Probability that bag contains exactly 3 black balls is 0.6 and probability of bag containing exactly 1 black ball is 0.4. Now, balls are drawn from the bag, one at a time, without replacement, till all black...

An electrical system consists of four components. The system works if components A and B work and either of the components C or D works. The reliability (probability of working) of each component is given by P(A) = 0.9, P(B) = 0.9, P(C) = 0.8, and P(D) = 0.8. Find the probability that (a) the...

Homework Statement We have 6 new and 4 used products in a box. We draw randomly 2 products and used them for a while. then return them back into the box. After that, again we draw 2 products. a) What is probability that at least one is new? b) What is probability that at least one is new if we...

Homework Statement Profesor has 17 keys, and only one unlocks the door. Let random variable X be attempt at which profesor unlocks the door (key that doesn't work is saved by side) Find: a) Most probable value of variable X b) In which attempt will profesor unlock the door c) Probability that...

Ok I am really stumped and have no clue what I am missing...Here is the scenario. We have a mother and father who can be either $BB$, $Bb$ or $bb$. The probability for both is:$$P(BB) = P(bb) = \frac{1}{4}$$ $$P(Bb) = \frac{1}{2}$$For the child one gene of the mother and one from the father is...

Hi, I am teaching a machine learning course and the students have very poor knowledge about conditional probability, Bayes rule etc. Most students have done their undergraduates years ago and I guess their educational background has not been that good. Last lecture was on Naive Bayes...

Homework Statement A company has been running a television advertisement for one of its new products. A survey was conducted. Based on its results, it was concluded that an individual buys the product with probability...

##P(A|A∩B) = \frac{P(A∩(A∩B))}{P(A∩B)} = \frac{P(A∩B)}{P(A∩B)} = 1## So given the the event "A and B" as the sample space, the probability of A occurring is 1. ##P(A|A∪B) = \frac{P(A∩(A∪B))}{P(A∪B)} = \frac{P(A)}{P(A∪B)}## Those two events are independent if and only if the probability of "A or...

P(A/B) is defined to be P(A∩B)/P(B) Why is this true? When A and B are dependent events, I can understand why this is correct. It is clear when you see the venn diagram. But for independent events, why is the formula correct? Any intuition or formal proof?

am a bit confused, if i want to find out for example the P(Having the disease among everyone) , using conditional, would it be total people Having the disease over total population?Prescreening Positive and Have the disease is 66 Prescreening Positive but does not the disease 150 prescreening...

Basically I am wondering how you deal with a conditional cdf and turning that into a conditional pdf when the random variables are independent. I know that f(X|Y) =f(X)f(Y)/f(Y)=f(X) I tried to derive this in a nice attached laTex document but it does not seem right to me. Note(this is for a...

Homework Statement Let X and Y be independent Bernoulli RV's with parameter p. Find, \mathbb{E}[X\vert 1_{\{X+Y=0\}}] and \mathbb{E}[Y\vert 1_{\{X+Y=0\}}] Homework EquationsThe Attempt at a Solution I'm trying to show that, \mathbb{E}[X+Y\vert 1_{\{X+Y=0\}}] = 0 by, \begin{align*}...

Homework Statement Let f(x1, x2, x3) = e-(x1+x2+x3), 0<x1,2,3<infinity, zero elsewhere be a joint pdf of X1, X2, X3. The variables are all independent to each other Compute P(X1< X2< X3|X3<1 ) Homework Equations P(X1< X2< X3|X3<1 ) The Attempt at a Solution P(X1< X2< X3|X3<1 )=P(X1< X2< X3<1...

So say I have Pr(Z|X&Y) I'm guessing that it follows the standard Pr(A|B)=[Pr(B|A)Pr(A)]/Pr(B) So Pr(Z|X&Y)=[Pr(X&Y|Z)Pr(Z)]/Pr(X&Y)?Also, if X&Y are independent, then would I get Pr(X&Y|Z)=Pr(X|Z)Pr(Y|Z)?

It is known that if it rains, there is a \(50\%\) chance that a sewer will overflow. Also, if the sewer overflows, then there is a \(30\%\) chance that the road will flood. If there is a \(20\%\) chance that it will rain, what is the probability that the road will flood? Let A be the...

Homework Statement Suppose there are two urns that contain white and yellow balls. Urn 1 contains 10 white and 5 yellow balls, and Urn 2 contains 6 white and 12 yellow balls. You are going to draw 3 balls without replacement from one of the urns. To decide which urn to draw from, you will...

I have a group of dogs (3 brown male, 2 brown female, 4 white male, 4 white female, 5 black male, 4 black female) What is the probability to 1. select a female brown dog ? 2. select a female, given that is a brown dog ? 3. select a brown given that is a female dog ? Thank you. I have...

Homework Statement We've got a standard deck of 52 cards. We shuffle the deck well, then cut it into two piles of 26 each, a top pile and a bottom pile. We reach in and pull a random card out of the top pile, observing that it is an Ace. We then put it into the bottom pile, shuffle the...

Dear All, I am a starter to machine learning and i am currently confused about the following problem: what is the result of P(X|Y)P(Y|Z)? In my book, it is written to be P(X|Z). But I don't think it is correct since P(X|Z)= P(X|Y,Z)P(Y|Z) But clearly P(X|Y)=/= P(X|Y,Z) Assuming...

I'd like some help understanding a proof, from http://www.statlect.com/cndprb1.htm. Properties are introduced, which a conditional probability ought to have: 1) Must satisfy properties of probability measures: a) for any event E, 0≤P(E)≤1; b) P(Ω)=1; c) Sigma-additivity: Let {E1, E2, ...

I really need some help here, will appreciate any effort. I calculated time series of tidal stresses. It turned out that the probability of having positive tidal stress is 0.4 and negative - 0.6 (I counted up number of hours when the stress was positive/negative and divided by the total number...

Hi guys, I am having trouble showing that \mathbb{E}\left[(Y-\mathbb{E}[Y|X])^{2}\right]=0. I understand the proof of why E[Y|X] minimizes the mean square error, but I cannot understand why it is then equal to zero. I tried multiplying out the square to get...

This is a fun question i found on the internet, a bit harder than my course and I've spent hours on it but can't find a solution, i was hoping someone could help me. Here's the situation. You are in jail and have been sentenced to death tomorrow, however there's a way out. you're given 12 red...

Homework Statement There are two types of cooking oil, mono- and polyunsaturated. In a supermarket, 10.526% of the oil sold is mono-, of this 3.684% is canola oil and 6.842% is corn oil. The remaining 89.48% of the oil sold is poly-, of this 48.95% is canola oil and 40.53% is corn oil...

Homework Statement A new diagnostic test is developed in order to detect a particular disease. It is known that 1% of the population has this disease. The diagnostic test is said to be 95% reliable. In other words, if a person has this disease, the test will detect it 95% of the time. On...

I can easily solve a problem if the probabilities are already defined in the question.. such as P(A|B) But I always get confused about how to actually define the probability from an application exercise. E.g.. Market research shows that (A)75% if customers want text messages on their phone...

w1,w2,w3,w4,w5 are normally distributed i.i.d R.V. I want to calculate the probability that slots 1&2(w1,w2) are smaller than slots 3,4,5 (where mu>0) I'm able to calculate the probability that slot 1 is smaller than 3,4,5, now I'm stuck on the calculation for slot 2, where i know that i...

Homework Statement A study of texting and driving has found that 40% of all fatal auto accidents are attributed to texting drivers, 1% of all auto accidents are fatal, and drivers who text while driving are responsible for 20% of all accidents. Find the percentage of non-fatal accidents...

Homework Statement The following experiment involves a single coin with probability p of heads on anyone flip, where 0 < p < 1. Step 1: Flip the coin. Let X = 1 if heads, 0 otherwise. Step 2: Flip the coin (X + 1) times. Let Y = the number of heads obtained in this step. Step 3: Flip the...

Hello all I have a little problem with this short question, would appreciate your help. Let A and B be two events such that: P(A|~B) = 1/2 and P(B?AUB) = 2/5 (~B means not B) Find P(B) The final answer should be 1/4, I can't get there. I did some work with the conditional probability...

Homework Statement Let X_1, X_2, X_3 be iid with common pdf f(x)=exp(-x), 0<x<infinity, 0 elsewhere. Evaluate P(X_1<X_2 | X_1<2X_2)Homework Equations f(X|Y) = f(x,y)/f(y) The Attempt at a Solution Since P(X_1<X_2) is a subset of P(X_1<2X_2), the intersection (edited, at first said union)...

Homework Statement Question 1: A teacher gave his class two tests where every student passed at least one test. 72% of the class passed passed both tests and 80% of the class passed the second test. (i)what percentage of those who passed the second test also passed the first? (ii) what...

I'm just going to post a screenshot of the Example (free online textbook). I'm having a tough time making the leap to the first sum - what allows me to rewrite P(T|A) as the sum of the product of those two conditional probabilities? Thanks

I'm casually working on determining the probability of a team in a given sport (let's say football) reaching at least a certain level in a season. There are two main parts to this: How many games they won in the season, and how far they got in the playoffs. I'd like to assign one final number...

Homework Statement From Mathematical Statistics and Data Analysis 3ed, Rice 1.8 #61 Suppose chips are tested and the probability they are detected if defective is 0.95, and the probability they are declared sound if they are sound is 0.97. If 0.005 of the chips are faulty. What is the...

Hello, I've been working on some questions from Rice's Mathematical Statistics and Data Analysis and I'm not sure about my solutions. The question I'm working on is as follows: 1.8 #59c A box has 3 coins: 1 with two heads, 1 with 2 tails and 1 fair coin. A coin chosen randomly is flipped...

Suppose that there are 2 apple trees. Tree A and tree B. A produces 0.7 of the farm's apples. And B produces 0.3. Out of the apples that tree A produces, 0.15 are bad. For B, 0.05 are bad. One package of goodies contains 3 apples. Given this information, what is the P(Tree A| at...

Hi, I was reading about Markov chains and came across the following statement: "The conditional distribution p(x_n|x_{n-1}) will be specified by a set of K-1 parameters for each of the K states of x_{n-1} giving a total of K(K-1) parameters." In the above we have assumed that the...

Homework Statement In 1988, the state of Illinois required HIV testing for a couple to obtain a marriage license. The HIV testing at the time consisted of two separate tests, the ELISA test and the Western Blot test. The Elisa test was signicantly less expensive. A person who is HIV...

Obviously, this is a homework assignment, so I don't want it done for me; however, I am confused. Perhaps I am just confused by the problem or the wording, but I am totally stuck on what to do. I believe the output signal should be a convolution where Z = X + Y, and Y is the gaussian(0, 2)...

Hi, I wish to confirm the results I obtained for the two following questions in statistics. I'd truly appreciate your feedback. Homework Statement 1) Die 1 and die 2 form a pair of unbiased dice. Die 1 has 4 faces painted red and 2 painted blue, whereas die 2 has 4 faces painted blue and 2...

It seems weird that such a relatively complex concept is simply given as a definition in most textbooks and then dismissed for further explanation other than using it intact or as a basis for further proofs.

Homework Statement A population has n men and n women. If you where to take 4 people out of the population to form a group what's the probability that there are exactly the same number of men as women in the group . Homework Equations The Attempt at a Solution Ok so I thought...

Okay so I have a complex setup that I hope I can convey. I have 9 sites to which X can bind. 6 out of the 9 sites are active and 3 out of the 9 sites are inactive. I need 3 of the active sites to be bound to get the response I am looking for - which we will call EMAX. So when I add a...