Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [conditional-probability]

For questions on conditional probability.

Conditional probability is the probability that an event occurs given that another event has already happened. The probability of an event $A$ given another event $B$ is written as $P(A|B)$, and is related to the marginal and joint probabilities via $$ P(A|B)P(B)=P(A\cap B)$$

5658 questions
0
votes
1 answer

Conditional Probability with One Complement

I'm given:$$x = .95 = P(A|B) = P(A^C|B^C)$$ $$P(B)=.05 \space \text{and} \space P(B^C)=.95$$ I want to find $\bf{P(B^C|A)}$ I know that: $$P(B^C|A) = \frac{P(B^C\cap A)}{P(A)}$$ I can find $P(A)$, but not sure what I can do do get me $P(B^C\cap…
brandon_ducks
  • 117
  • 7
0
votes
2 answers

Explain dependency between probability event

Say I have 7 cards that are numbered from 1 to 7. $D = \{1,2,3,4,5,6,7\}$ and events A and B: $A =:$ getting an even number $\rightarrow \{2,4,6\}$ $B =:$ getting a number bigger than $4 \rightarrow \{5,6,7\}$ Then it follows: $P(A) = 3/7…
learning_dude
  • 101
0
votes
2 answers

How should I understand conditional probability?

I think now everyone know the formula for calculating conditional probability $$P(B|A) = \frac{P(A \bigcap B)}{P(A)}$$ But I'm having a hard time understanding it via a problem. Let say we have 3 marble, 1 red, 1 blue, 1 yellow. Now intuitively I…
Airi
  • 3
0
votes
1 answer

Conditioning in total law of probability

Conditioning on other random variables is something I don't have a great grasp of. I came across example 3.2 in 'Random Processes for Engineers' by professor hajek and this concept seems to be what prevents me from finishing off the example. In…
Bolpincal
  • 33
0
votes
1 answer

What is the posterior probability that the red ball came from box $1$?

There are two boxes. Box $1$ contains three red and five white balls and box 2 contains two red and five white balls. A box is chosen at random $p(box = 1) = p(box = 2) = 0.5$ and a ball chosen at random from this box turns out to be red. What is…
Slim Shady
  • 363
0
votes
0 answers

Conditional probability - 2 students

I am having difficulty grasping conditional probability. This seemingly innocuous problem, or rather its solution, is confusing me: The probability of student A solving an assignment is 0.7 and the probability of student B solving an assignment is…
Tamara Todorovic
  • 11
0
votes
1 answer

Find the conditional distribution of X=1 given Z=1

Let $X$ follows Bernoulli(1/3), $Y$ independent of $X$ follows Bernoulli(2/3). $$Z=\begin{cases}X &\text{if $Y=1$}\\ 1-X &\text{if $Y=0$} \end{cases}$$ Find the conditional distribution of $X=1$ given $Z=1$. I was applying ${P(X=1,Z=1)\over…
Nandita Mukherjee
  • 19
0
votes
1 answer

Averaging conditional density over an event

I expect this should be a basic property of regular conditional densities/stochastic kernels, but somehow I am having trouble verifying this. Suppose we have random variables $X$ and $Y$ with (smooth) joint probability density $p(x,y)$ and (smooth)…
pseudocydonia
  • 2,144
0
votes
2 answers

Probability Confusion/Clarification

Two pizzas are each divided into eight slices and placed inside separate boxes with the lids closed, and left in an empty room. People randomly arrive at the room to take a slice. Each person who arrives randomly chooses a box, opens it, takes a…
RedG
  • 267
0
votes
1 answer

Probability of an event given conditional and unconditional probabilities

Let's assume the following: a) the conditional probability of $B$ given $A$ is 0.8 b) the conditional probability of $B$ given $\text{not }A$ is 0.4 c) the unconditional probability of $B$ is 0.5 What is the probability of $A$? EDIT I'm really…
equanimity
  • 123
0
votes
0 answers

Conditional probability - Prove or disprove

For all discrete probabilities $P$ and all events $X$ and $Y$, prove or disprove: If $0 < P(X) \leq P(Y)$, then $P(X|Y) \leq P(Y|X)$. My attempt goes something like this (which I am sure is wrong): Consider $X$ and $Y$ are independent of each…
user858376
0
votes
1 answer

Expanding $P(Z \cap X \mid Y)$

I can't see exactly why $P(Z \cap X \mid Y) = P(Z \mid X \cap Y) \mathop{P}(X \mid Y)$. I have been told (and it is also my intuition) that this has to do with the definition of conditional probability, but I can't see how this works in detail. Can…
Mijito
  • 235
0
votes
0 answers

An exercise on Conditional probability

A family has two children. Assume that birth month is independent of gender, with boys and girls equally likely and all months equally likely, and assume that the elder child’s characteristics are independent of the younger child’s…
xuhui zeng
  • 1
0
votes
0 answers

Probability difference in playing the same numbers every time

I joined a Lotto pool of 3 people four months ago. Since then, we've pooled \$200 once a month for the last four consecutive months, to buy 100 games in the \$2 Cash4Life game. Odds on the jackpot are 21 million:1, but 210,000:1 with 100 tickets.…
Tom Cullem
  • 19
0
votes
0 answers

Conditional probability of $Y > 250$ given $X$ and $Y$ is either greater than $250$ or it's not.

Suppose $Y=\beta_1 X + \beta_2 X^2$ for some real numbers $\beta_1$ and $\beta_2$ where $X$ is a random variable with real values greater than $0$ and $Y$ is greater than $0$. What is the probability $P[Y>250 | X]$ if you consider $Y$ in a binary…
E2R0NS
  • 256
1 2 3
…
15 16