Kimi | Mean & Variance of Multiple Fs

When given the same input item $B$ processes each return either T or F. The probability that process $S_i$, $1 <= i <= B$, returns T is $p_i$, and this probability is independent of any other process. What is the expected probability that two or more processes return F for the same input item?

This is a probability problem involving independent Bernoulli trials. Let me analyze it step by step.

Understanding the Problem

We have

B

independent processes, where:

Process $S_{i}$ returns T with probability $p_{i}$
Process $S_{i}$ returns F with probability $q_{i} = 1 - p_{i}$

We want the expected probability that two or more processes return F for the same input.

Wait - since the probabilities

p_{i}

are fixed (not random variables), the probability we're looking for is actually a deterministic value, not an expectation in the statistical sense. The phrase "expected probability" likely just means "what is the probability."

Solution Approach

It's easier to use the complement rule:

P (2 or more F’s) = 1 - P (0 F’s) - P (exactly 1 F)

Step 1: Probability of 0 F's (all T's)

P (all T) = \prod_{i = 1}^{B} p_{i}

Step 2: Probability of exactly 1 F

This means exactly one process returns F and the rest return T:

P (exactly 1 F) = \sum_{j = 1}^{B} q_{j} \prod_{i \neq = j} p_{i} = \sum_{j = 1}^{B} \frac{q _{j}}{p _{j}} \prod_{i = 1}^{B} p_{i}

Or equivalently:

P (exactly 1 F) = (\prod_{i = 1}^{B} p_{i}) \sum_{j = 1}^{B} \frac{1 - p _{j}}{p _{j}}

Step 3: Final Answer

P (2 or more F’s) = 1 - i = 1 \prod B p_{i} - j = 1 \sum B q_{j} i \neq = j \prod p_{i}

Or in expanded form:

= 1 - \prod_{i = 1}^{B} p_{i} (1 + \sum_{j = 1}^{B} \frac{1 - p _{j}}{p _{j}})

Alternative Expression

If we define

Q = \prod_{i = 1}^{B} p_{i}

, then:

P (2 or more F’s) = 1 - Q - Q \sum_{j = 1}^{B} \frac{q _{j}}{p _{j}} = 1 - Q (1 + \sum_{j = 1}^{B} \frac{q _{j}}{p _{j}})

Special Case: Equal Probabilities

If all

p_{i} = p

(so

q_{i} = q = 1 - p

for all

i

$P (0 F’s) = p^{B}$
$P (exactly 1 F) = (1 B) q p^{B - 1} = Bq p^{B - 1}$

P (2 or more F’s) = 1 - p^{B} - Bq p^{B - 1} = 1 - p^{B - 1} (p + Bq)

Or using the binomial distribution directly:

= \sum_{k = 2}^{B} (k B) q^{k} p^{B - k}

This is the standard binomial tail probability.

When $N$ independent input items have been processed, what is the mean and variance of the number of input items where at least two processes returned F?

I need to find the mean and variance of the number of items (out of

N

independent items) where at least two processes return F.

Setup

Let me define for each input item

n

(where

n = 1, 2, \dots, N

$X_{n} = 1$ if at least 2 processes return F for item $n$ , and $X_{n} = 0$ otherwise.

The total count is:

Y = \sum_{n = 1}^{N} X_{n}

Finding the Distribution of $X_{n}$

From the previous problem, for a single item:

p = P (X_{n} = 1) = P (at least 2 F’s) = 1 - \prod_{i = 1}^{B} p_{i} - \sum_{j = 1}^{B} q_{j} \prod_{i \neq = j} p_{i}

where

q_{j} = 1 - p_{j}

X_{n} \sim Bernoulli (p)

, and since items are independent:

Y \sim Binomial (N, p)

Mean of Y

For a Binomial distribution:

E [Y] = Np = N 1 - i = 1 \prod B p_{i} - j = 1 \sum B q_{j} i \neq = j \prod p_{i}

Variance of Y

For a Binomial distribution:

Var (Y) = Np (1 - p) = N 1 - i = 1 \prod B p_{i} - j = 1 \sum B q_{j} i \neq = j \prod p_{i} i = 1 \prod B p_{i} + j = 1 \sum B q_{j} i \neq = j \prod p_{i}

Simplified Notation

Let me define:

$P_{0} = \prod_{i = 1}^{B} p_{i}$ (probability all T)
$P_{1} = \sum_{j = 1}^{B} q_{j} \prod_{i \neq = j} p_{i}$ (probability exactly 1 F)

Then:

$p = 1 - P_{0} - P_{1}$ (probability of at least 2 F's)
$1 - p = P_{0} + P_{1}$ (probability of 0 or 1 F)

So:

E [Y] = N (1 - P_{0} - P_{1})

Var (Y) = N (1 - P_{0} - P_{1}) (P_{0} + P_{1})

Special Case: Equal Probabilities

If all

p_{i} = p

and

q = 1 - p

$P_{0} = p^{B}$
$P_{1} = Bq p^{B - 1}$

Then:

E [Y] = N (1 - p^{B} - Bq p^{B - 1}) = N (1 - p^{B - 1} (p + Bq))

Var (Y) = N (1 - p^{B} - Bq p^{B - 1}) (p^{B} + Bq p^{B - 1})

List methods of obtaining confidence intervals when $p_i$ is very close to 1

When

p_{i} \approx 1

(so

q_{i} = 1 - p_{i} \approx 0

), we're dealing with rare events (few F's). This is the "high reliability" regime where failures are uncommon. Here are the main methods for constructing confidence intervals in this setting:

1. Standard Wald Interval (Often Poor Here)

\overset{p}{^} \pm z_{α /2} \frac{p ^ ( 1 - p ^ )}{N}

Problem: When

p \approx 1

, if we observe few failures,

\overset{p}{^}

can be exactly 1, giving zero-width intervals. The normal approximation fails for extreme probabilities with moderate

N

2. Wilson Score Interval (Better)

\frac{p ^ + \frac{z ^{2}}{2 N} \pm z \frac{p ^ ( 1 - p ^ )}{N} + \frac{z ^{2}}{4 N ^{2}}}{1 + \frac{z ^{2}}{N}}

Works better than Wald for extreme $p$
Always gives intervals inside $[0, 1]$
Good for $p \approx 1$ with reasonable $N$

3. Clopper-Pearson (Exact) Interval

Based on the binomial CDF, solving:

\sum_{k = y}^{N} (k N) p_{L}^{k} (1 - p_{L})^{N - k} = α /2

\sum_{k = 0}^{y} (k N) p_{U}^{k} (1 - p_{U})^{N - k} = α /2

Guarantees coverage probability $\geq 1 - α$
Conservative (wider than necessary)
Computationally intensive but reliable

4. Agresti-Coull Interval (Adjusted Wald)

Add "pseudo-observations":

\tilde{N} = N + z^{2}

\tilde{y} = y + z^{2} /2

\tilde{p} \pm z \frac{p ~ ( 1 - p ~ )}{N ~}

Better coverage than Wald near boundaries
Simple to compute

5. Bayesian Credible Intervals (Highly Recommended)

Beta prior + Binomial likelihood → Beta posterior

For

p \approx 1

, use Jeffreys prior:

Beta (1/2, 1/2)

or informative prior concentrated near 1.

Posterior:

Beta (y + 1/2, N - y + 1/2)

Credible interval from quantiles of Beta distribution.

6. Poisson Approximation (For Very Rare F's)

When

p_{i} \approx 1

q_{i} \approx 0

, and

N

is large:

Y =

count of items with

\geq 2

F's, and

p = P (\geq 2 F’s) \approx 0

Y \approx Poisson (λ = Np)

Confidence interval for

λ

λ_{L}, λ_{U}

from Poisson tables, then scale back.

7. Logit Transformation (Variance Stabilizing)

logit (\overset{p}{^}) = ln (\frac{p ^}{1 - p ^}) \approx N (logit (p), \frac{1}{Np ( 1 - p )})

CI for

logit (p)

, then transform back. Better for

p \approx 0

1

8. Bootstrap Methods

Percentile bootstrap: Resample data, compute $\overset{p}{^}^{*}$ for each sample, use percentiles
BCa (bias-corrected accelerated): More accurate, adjusts for skewness

9. Likelihood Ratio Interval

Solve:

- 2 ln (\frac{L ( p )}{L ( p ^ )}) \leq χ_{1}^{2} (α)

Where

L (p) = p^{y} (1 - p)^{N - y}

. More accurate than Wald, especially near boundaries.

10. Rule of Three (Conservative, Quick)

If you observe

y = 0

events in

N

trials:

p_{U} \approx \frac{3}{N}

For one-sided upper bound on rare event rate.

Recommendations for $p_{i} \approx 1$

表格

Scenario	Recommended Method
Small $N$ ( $< 30$ )	Clopper-Pearson or Bayesian with Beta prior
Moderate $N$	Wilson score or Agresti-Coull
Large $N$ , very rare failures	Poisson approximation
Need exact coverage	Clopper-Pearson
Prior information available	Bayesian credible intervals
Computational flexibility	Bootstrap (BCa preferred)

Special Note for Your Problem

When

p_{i} \approx 1

, the probability

p = P (\geq 2 F’s)

involves products of small

q_{i}

's:

$P_{0} = \prod p_{i} \approx 1 - \sum q_{i}$ (very close to 1)
$P_{1} = \sum q_{j} \prod_{i \neq = j} p_{i} \approx \sum q_{j}$ (small)
$p = 1 - P_{0} - P_{1} \approx 0$ (very small!)

So you're estimating a very small probability from binomial data. The Poisson approximation or Bayesian methods with informative priors are particularly valuable here.

Mean & Variance of Multiple Fs

Understanding the Problem

Solution Approach

Step 1: Probability of 0 F's (all T's)

Step 2: Probability of exactly 1 F

Step 3: Final Answer

Alternative Expression

Special Case: Equal Probabilities

Setup

Finding the Distribution of Xn​

Mean of Y

Variance of Y

Simplified Notation

Special Case: Equal Probabilities

1. Standard Wald Interval (Often Poor Here)

2. Wilson Score Interval (Better)

3. Clopper-Pearson (Exact) Interval

4. Agresti-Coull Interval (Adjusted Wald)

5. Bayesian Credible Intervals (Highly Recommended)

6. Poisson Approximation (For Very Rare F's)

7. Logit Transformation (Variance Stabilizing)

8. Bootstrap Methods

9. Likelihood Ratio Interval

10. Rule of Three (Conservative, Quick)

Recommendations for pi​≈1

Special Note for Your Problem

Finding the Distribution of $X_{n}$

Recommendations for $p_{i} \approx 1$