What is Conditional probability: Definition and 242 Discussions
In probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(A|B), or sometimes PB(A) or P(A/B). For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person is sick, then they are much more likely to be coughing. For example, the conditional probability that someone unwell is coughing might be 75%, in which case we would have that P(Cough) = 5% and P(Cough|Sick) = 75%.
Conditional probability is one of the most important and fundamental concepts in probability theory. But conditional probabilities can be quite slippery and might require careful interpretation. For example, there need not be a causal relationship between A and B, and they don't have to occur simultaneously.
P(A|B) may or may not be equal to P(A) (the unconditional probability of A). If P(A|B) = P(A), then events A and B are said to be independent: in such a case, knowledge about either event does not alter the likelihood of each other. P(A|B) (the conditional probability of A given B) typically differs from P(B|A). For example, if a person has dengue, they might have a 90% chance of testing positive for dengue. In this case, what is being measured is that if event B ("having dengue") has occurred, the probability of A (test is positive) given that B (having dengue) occurred is 90%: that is, P(A|B) = 90%. Alternatively, if a person tests positive for dengue, they may have only a 15% chance of actually having this rare disease, because the false positive rate for the test may be high. In this case, what is being measured is the probability of the event B (having dengue) given that the event A (test is positive) has occurred: P(B|A) = 15%. Falsely equating the two probabilities can lead to various errors of reasoning such as the base rate fallacy. Conditional probabilities can be reversed using Bayes' theorem.
Conditional probabilities can be displayed in a conditional probability table.
using the equation mentioned under Relevant Equations I can get, $$\mathbb{P}(2X > Y |1 < 4Z < 3) = \frac{\mathbb{P}(2X>Y, 1<4z<3)}{\mathbb{P}(1<4z<3)}$$ I can find the denominator by finding the marginal probability distribution, ##f_{Z}(z)## and then integrating that with bounds 0 to 1. But I...
I submitted this solution, and it was marked incorrect. Could I get some feedback on where I went wrong?
Let S represent the event that Party A wins the senate and H represent the event that Party A wins the house.
There are 4 cases: winning the senate and house (##S \cap H##), winning just...
North and south have ten trumps between them ( trumps being cards of specified suit).
(a) Find the probability that all three remaining trumps are in the same hand. (that is either east or west has no trumps).
(b) If it is known that king of trumps is included among the three, what is the...
Problem: In a dresser there are 3 drawers. In one drawer there are two black socks and one white sock, in the second drawer there are two white socks, and in the third drawer there is a black and white sock. Suppose I chose a drawer randomly ( meaning, in a uniform distribution ) and I took a...
My attempt:
$$P(\text{B is positive}|\text{A is positive})=\frac{P(\text{B is positive} \cap \text{A is positive})}{P(\text{A is positive})}$$
$$=\frac{P(\text{B is positive})\times P(\text{A is positive})}{P(\text{A is positive})}$$
$$=P(\text{B is positive})$$
$$=0.01 \times 0.99 + 0.99 \times...
Hello,
I have a question about the following sentence and would appreciate if someone could explain how to read out the conditional probability here.
"Each microwave produced at factory A is defective with probability 0.05".
I understand the sentence as the intersection ##P(Defect \cap...
We know that ##P(A-) = (95\% \cdot 0.5\% + 5\% \cdot 98.5\% )## and ##P(guilty \ and \ A-) = (95\% \cdot 0.5\%)##, so letter a) is just ##P(guilty \ and \ A-)/P(A-)##.
What I tried to do in letter b) was again using the conditional probability theorem. First calculating the probability that...
For letter a), i think that he is assuming that each hypothesis is independent, and that they are mutually exclusive.For letter b), I understand that it indeed admits the relative frequency interpretation, since the the experiment is being produced several times.
For letter c) we do the...
Hi,
Just a quick question about conditional and marginal probabilities notation.
Question: What does ## p(a|b, c) ## mean?
Does it mean:
1) The probability of A, given (B and C) - i.e. ## p[A | (B \cap C)] ## OR
2) The probability of (A given B) and C - i.e. ## p[(A | B) \cap C] ##
I was...
Confused and not sure if it is correct, but please do correct my steps.
We let event B be that there are at least 3 customers entering in 5 minutes.
Hence P(B) = 1- P(X=0)- P(X=1) - P(X=2) = ##1- \dfrac{e^{-5}5^{0}}{0!}-\dfrac{e^{-5}5^{1}}{1!}-\dfrac{e^{-5}5^{2}}{2!} ## = 0.8753...
Now we let...
Summary:: There's 11 fruits, 3 of which is poisionous.
A guy eats 4 of them, a girl eats 6 and a dog gets the last one.
What is the conditional probability of both the girl and guy dying IF the dog made it? One fruit is enough to kill you.
$$P(dog lives) = 8/11$$
$$P(allPeopleDie | dog...
1. Definition
If E and F are two events associated with the same sample space of a random experment, the conditional probability of the event E given that F has occurred, i.e. P(E|F) is given by
P(E|F) = (E∩F)/P(F) (P≠0)
2. Properties of conditional probability
Let E and F be events of...
I am a noob to this topic so correct me If I made any silly mistake. By plugging in the values I managed to get
p(abc)=0.75*0.9*p(c|ab)
Here How can I find p(c|ab)? Is this question unsolvable or can I derive it?
I also want to know what is meant by p(abc) in literary terms.
I also created a...
Question: "In a lottery game each player tries to guess right 6 numbers designated in advance by choosing randomly from among numbers from 1 to 20. Given that one player guessed right 5 numbers out of 6 that he/she picked, what is the probability of guessing right the 6 numbers?"
The problem...
##P(T=1|W=w)=\frac{P(\{T=1\}\cap\{W=w\})}{P(W=w)}=\frac{\binom {n-2} {w-1} p^{w-1}(1-p)^{(n-2)-(w-1)}}{\binom n w p^w (1-p)^{n-w}}=\frac{(n-2)!}{(w-1)!(n-w-1)!}\frac{w!(n-w)!}{n!}\frac{1}{p(1-p)}=\frac{w(n-w)}{n(n-1)}(p(1-p))^{-1}##.
I cannot seem to get the terms with ##p## out of my expression.
There are 4 books being sold in the bookshop : A, B, C, D.
We know that 20% of the male customers buy book A at least once a week, 55% buy book B at least once a week, 25% buy book C at least once a week and 15% buy book D at least once in a month.
We also know that 32% of the female customers...
For 1) I found two ways but I get difference results.
The first way is I use P(A|B) = P(A and B)/P(B).
I get P(X<1|Y<1)=(∫_0^1▒∫_0^1▒〖3/4 (2-x-y)dydx〗)/(∫_0^1▒∫_0^1▒〖3/4 (2-x-y)dydx〗+∫_1^2▒∫_0^(2-x)▒〖3/4 (2-x-y)dydx〗)=6/7
The 2nd method is I use is
f(x│y)=f(x,y)/(f_X (x)...
Hello!
I am trying to get to grips with the Bayes' formula by developing an intuition about the formula itself, and on how to use it, and how to interpret.
Please, take a problem, and my questions written within them - I will highlight my questions and will post them as I add the information...
Hello.
I am reading an online stats book, and there is the following question, which I solved incorrectly, and I think I understand what is my mistake, but I will be grateful for your explanation, if I have incorrectly detected the logic behind my mistake. I am weak at math (trying to improve it...
I am having a hard time with the following exercise:
Assume for this problem that the company has 8 Chevrolets and 4 Jeeps, and two cars are selected randomly and given to sales representatives.
What is the probability of both cars being Chevrolets, given that both are of the same make?
I...
Hey! I need help with my Math homework :( The question is the following...
There are 5 history courses of interest to Howard, including 3 in the afternoon, and there are 6 psychology courses, including 4 in the afternoon. Howard picks a course by selecting a dept at random, then selecting a...
Homework Statement
Out of all the products a company makes 2% is damaged. During the routine control of the products, the products are put to a test which discovers the damaged ones in 99% of the cases. In 1% however it approves the damaged item as a working one and vice versa. Find the...
Homework Statement
what is the probability that a component which is still working after 800 hrs, will last for at least 900hrs
Homework Equations
conditional probability
P(E|A) = ( P( E ∩ A) ) / ( P(A) )
The Attempt at a SolutionIm just checking my own understanding if this problem is...
Hi,
I was having some trouble doing some bayesian probability problems and was wondering if I could get any help. I think I was able to get the first two but am confused on the last. If someone could please check my work to make sure I am correct and help me on the last question that would be...
I'm currently stuck on a question that involves conditional probability with 3 events. This is a concept that I'm having the most trouble grasping and trying to solve in this subject. I am not sure how to start this problem.
The Question:
Given that P(A n B) = 0.4, P(A n C) = 0.2, P(B|A)=0.6...
Hi
In Dudas Pattern Classification, he Writes that P(x,\theta|D) can always be written as P(x|\theta,D)P(\theta|D) . However, I cannot find any justification for this. So, why are these Equal?
I don't get $$\frac{P[x<X<x+dx|N=n]}{dx}=f_{X|N}(x|n)$$ Can someone derive why? I would believe that $$f_{X|N}(x|n)=\frac{f(x,N)}{p_n(N)}$$ but I don't get how that would be the same. And I don't get that $$\frac{P[x<X<x+dx|N=n]}{dx}=\frac{P[N=n|x<X<x+dx]}{P[N=n]}\frac{P[x<X<x+dx]}{dx}$$
Can...
(Mentor note: link removed as not essential to the question.)
The problem is: what is relevance anyhow?
My questions are these: did I get the math right in the following? Is there a better, more acceptable way to lay out the sample space Ω and the two events F and E? Apart from the math...
The theorem says
The probability that an event B occur after A has already occurred is given by
P(B/A) =P(A intersection B) /P(A)
But applying thus to a problem like the probability of occurrence of all 3 tails on 3 coins when tossed if 1 tail has already occurred is
P(B/A) =(1/8)/(7/8)=1/7...
This question has been driving me crazy.
A large industrial firm uses three local motels to provide overnight accommodations for its clients.
From past experience it is known that 20% of the clients are assigned rooms at the Ramada Inn, 50% at the Sheraton and 30% at Lakeview. What is the...
Hey! :o
Let $P$ be a probability measure on a $\sigma$-Algebra $\mathcal{A}$. I want to prove or disprove the following statements:
$P(A\mid B)=1-P(\overline{A}\mid B)$, for $A, B\in \mathcal{A}$
$P(A\mid B)=1-P(A\mid \overline{B})$, for $A, B\in \mathcal{A}$
I have done the following...
I scored in the 88th percentile in a certain personality trait and am trying to figure out the probability of that given that I'm male. I'm trying the likelihood that I would land in the 88th percentile given that I'm male.
Definitions: T = trait, M = males, F = female.
Given:
P(T|M) = 0.3...
Su, Francis, et. al. have a short description of the paradox here: https://www.math.hmc.edu/funfacts/ffiles/20001.6-8.shtmlI used that link because it concisely sets forth the paradox both in the basic setting but also given the version where the two envelopes contain ( \,\$2^k, \$2^{k+1}) \...
Dear All sorry for repeated post;
There is a problem
Problem: Three cards are drawn in succession from a deck without replacement. find the probability distribution for the number of spades.
I have come with this solution.
Let S1: appearance of spade on first draw S2: appearance of spade on 2nd...
Dear all Please help in solving the following problem.
A large industrial firm uses 3 local motels to provide overnight accommodations for its clients. from past experience, it is known that 20% of the clients are assigned rooms at the Ramada Inn, 50% at the sheeraton and 30% at the Lakeview...
Hi guys,
I have a question about computing conditional probabilities of a Poisson distribution.
Say we have a Poisson distribution P(X = x) = e^(−λ)(λx)/(x!) where X is some event.
My question is how would we compute P(X > x1 | X > x2), or more specifically P(X> x1 ∩ X > x2) with x1 > x2?
I...
Homework Statement
It rains in a city with a chance of 0.4. The weather forecast is not always accurate. When there will be a rain the next day, the forecast predicts the rain with probability 0.8; When there is no rain, the forecast falsely predicts a rain with probability 0.1. You take your...
Need help with a probability problem. I have the answer from the answer key, I just don't know how to figure it out.An insurance company examines its pool of auto insurance customers and gathers the following information:1) All customers insure at least one car.
2) 70% of the customers insure...
Homework Statement
The probability density function for a random vector ##(X,Y)## is ##f(x,y) = 3x##, when ##0 < y< x < 1##. Calculate the conditional probability
P(X> \frac{1}{2} | Y > \frac{1}{3})
Homework Equations
Conditional probability:
\begin{equation}
P(A | B) = \frac{P(A \cap...
Hey there community, I have a question on an exercise. Actually it is a general question based on it. Here is the exercise:
We throw 3 dice. If we know that the sum of these 3 is 10, then what is the probability of at least one of them being 3?
Well now, this exercise is very simple. I mean I...
I've found this video about conditional probability:
All steps look correctly, but the result does not make any sense.
I'm ok with the part about frogs, but not so with the boy/girl computation.
To sum it up:
1) I have two children and at least one of them is a boy. What is a probability I...
Homework Statement
suppose we have 9 balls : 2 red, 3 green, 4 yellow. and we draw 2 balls without replacement, the probability that one of them is red and the other is green is : P(R)P(G\R)+P(G)P(R\G) = (2/9)(3/8)+(3/9)(2/8)
i faced a problem in the textbook which says: the probability that a...
As stated in my subject line, I know that P(A|B) = P(A) and P(B|A) = P(B), i.e. A and B are separable as P(A,B) = P(A) P(B). I strongly suspect that this holds with a conditional added, but I can't find a way to formally prove it... can anyone prove this in a couple of lines via Bayes' rules...
Hi,
I found this screenshot on a website and I thought it was crazy. I want to calculate the conditional probability of this event occurring because it seems so impossible.
Assume NLTH is being played. I want to calculate the conditional probability of this hand being dealt. Here is a...
Homework Statement
The problem statement is given below:
Homework EquationsThe Attempt at a Solution
Here is my attempt so far:
I'm sure questions 1 - 4 have been answered. Question 5 is what concerns me.
I need to find ##P(C' | D')##, which is the probability a good item is...
Homework Statement
Determine ##P(X<Y|x>0)##
Homework Equations
X and Y are random variables with the joint density function
$$
f_{XY}(x,y)=
\begin{cases}
4|xy|,-y<x<y,0<y<1\\
0,elsewhere
\end{cases}$$
The marginal densities are given by
$$
f_X(x)=2x\\
f_Y(y)=4y^3
$$
The Attempt at a Solution...