Intereting Posts

Number of nonnegative integer solutions to $x_1+x_2+x_3\le10$ with $x_1 \ge 1\ ,\ x_2\ge3$
Is it possible to 'split' coin flipping 3 ways?
“Fair” game in Williams
subgroup generated by two subgroups
The last few digits of $0^0$ are $\ldots0000000001$ (according to WolframAlpha).
How to calculate the asymptotic expansion of $\sum \sqrt{k}$?
Question on the fill-in morphism in a triangulated category
Variant of the Vitali Covering Lemma
Is there an elementary way to see that there is only one complex manifold structure on $R^2$?
Characteristics method applied to the PDE $u_x^2 + u_y^2=u$
Is the matrix $A^*A$ and $AA^*$ hermitian?
Symmetrize eigenvectors of degenerate (repeated) eigenvalue
Differentiation under integral sign (Gamma function)
Continuous functions do not necessarily map closed sets to closed sets
Distance/Similarity between two matrices

A and B are independent witness in a case. The probablity that A

speaks the truth is ‘x’ and that of B is ‘y’.If A and B agree on a

certain statement, how to find the probability that the statement is

true ?

- How to choose between two options with a biased coin
- Probability of iid random variables to be equal?
- A variant of the Monty Hall problem
- Does Convergence in probability implies convergence of the mean?
- Given a set of partial orderings of samples from a set of distributions, can we estimate the (relative) mean of the distributions?
- Why is the expected number coin tosses to get $HTH$ is $10$?
- What is an alternative book to oksendal's stochastic differential equation: An introduction?
- The king comes from a family of 2 children. What is the probability that the other child is his sister?
- Bayesian Parameter Estimation - Parameters and Data Jointly Continuous?
- What's the probability of a an outcome after N trials, if you stop trying once you're “successful”?

Let:

$A_t$ stand for “A says statement is true.” and $A_f$ for “A says statement is false” and

$B_t$ stand for “B says statement is true.” and $B_f$ for “B says statement is false” and

$S_t$ stand for “Statement is true” and $S_f$ for “Statement is false” and

Then, we know that:

$\text{Prob}(A_t | S_t) = \text{Prob}(A_f | S_f) = x$, and

$\text{Prob}(A_t | S_f) = \text{Prob}(A_f | S_t) = 1-x$, and

$\text{Prob}(B_t | S_t) = \text{Prob}(B_f | S_f) = y$, and

$\text{Prob}(B_t | S_f) = \text{Prob}(B_f | S_t) = 1-y$, and

We want to know:

$\text{Prob}(S_t | A_t \cap B_t)$

Using Bayes theorem, we have:

$$\text{Prob}(S_t | A_t \cap B_t) = \frac{\text{Prob}(A_t \cap B_t |S_t) \text{Prob}(S_t)}{\text{Prob}(A_t \cap B_t)}$$

But, we know that,

$\text{Prob}(A_t \cap B_t |S_t) = xy$ and

$\text{Prob}(A_t \cap B_t) = \text{Prob}(A_t \cap B_t |S_t) \text{Prob}(S_t) + \text{Prob}(A_t \cap B_t |S_f) \text{Prob}(S_f)$

Thus,

$\text{Prob}(A_t \cap B_t) = xy \text{Prob}(S_t) + (1-x)(1-y) (1-\text{Prob}(S_t))$

Simplifying the above, we get:

$\text{Prob}(A_t \cap B_t) = (1+2xy-x-y) \text{Prob}(S_t)$

Thus, we have:

$$\text{Prob}(S_t | A_t \cap B_t) = \frac{xy}{1+2xy-x-y}$$

$ P(A)=x $ and $ P(B)=y $,

$A$ and $B$ are independent, so $P(A \cap B)=P(A)P(B)$

Therefore, the probability that both speak the truth will be $P(A \cap B)=P(A)P(B)=xy$.

And then, the probability that they agree on a certain statement is, they both speak the truth or they both tell lie, which would be:

$$xy + (1-x)(1-y) $$

As a result, the probability that the statement is true is:

$$ \frac{xy}{xy+(1-x)(1-y)} $$.

Say the fact is True or False (T/F), and independent statements A, B are binary.

We are given P(A=1|T)=x and P(B=1|T)=y.

By convexity P(A=0|T)=1-x and P(B=0|T)=1-y.

We can surmise P(A=0|F)=x and P(B=0|F)=y.

Also by convexity P(A=1|F)=1-x and P(B=1|F)=1-y.

We want P(T|A=1,B=1).

By Bayes rule P(T|A=1,B=1) = P(A=1,B=1|T)P(T)/P(A=1,B=1).

By independence P(T|A=1,B=1) = P(A=1|T)P(B=1|T)P(T)/(P(A=1)P(B=1)).

Plugging in, the numerator is xyP(T).

The denominator requires P(A=1) = P(A=1|T)P(T) + P(A=1|F)P(F) = xP(T) + (1-x)P(F), same for P(B=1).

So I don’t think we are identified without knowing the marginal probability of the truth P(T).

Of course, for a given x and y you can provide P(T|A=1,B=1) as a function of P(T) between 0 and 1.

- Transitive subgroup of symmetric group
- Is the empty set an open ball in a metric space?
- Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
- Estimating $\sum n^{-1/2}$
- Solving a second-order linear ODE: $\frac{d^2 y}{dx^2}+(x+1)\cdot \frac{dy}{dx}+5x^2\cdot y=0$
- Secret santa problem
- Proof concerning Mersenne primes
- Prove a set of nonnegative integers with greatest common divisor 1 and closed under addition has all but finite many nonnegative integers.
- Inner Product on $\mathbb{R}$ and on $\mathbb{C}$
- Checking of a solution to How to show that $\lim \sup a_nb_n=ab$
- An infinite subset of a countable set is countable
- How do I show that $\int_{-\infty}^\infty \frac{ \sin x \sin nx}{x^2} \ dx = \pi$?
- How to determine the difference Onto vs One-to-one?
- Proof that the sum of the even side and the hypotenuse of a coprime (and positive) Pythagorean triple is a square number
- Let $0<x<1$ and $p<1$ then can I conclude the following : $(1-x)^p<1-px$