Raised Fist0

Bayes’ Theorem

Start learning this pattern below

Jump into concepts and practice - no test required

or
Recommended
Test this pattern10 questions across easy, medium, and hard to know if this pattern is strong

Introduction

Bayes’ Theorem हमें किसी नई information या evidence मिलने पर probabilities को revise करने में मदद करता है। यह real-world applications जैसे medical diagnosis, machine learning और decision-making में बहुत उपयोगी है, जहाँ किसी घटना की probability को observed data के आधार पर update करना होता है।

Pattern: Bayes’ Theorem

Pattern: Bayes’ Theorem

Bayes’ Theorem event A की probability को update करता है जब event B observe हो चुका हो।

Formula:
P(A | B) = [P(B | A) × P(A)] / [P(B | A)P(A) + P(B | A′)P(A′)]

Step-by-Step Example

Question

दो machines, M₁ और M₂, bulbs बनाती हैं। M₁ कुल bulbs का 40% बनाती है और उसका defect rate 2% है। M₂ कुल bulbs का 60% बनाती है और उसका defect rate 3% है। यदि एक bulb defective निकला, तो उसकी probability क्या है कि वह M₁ द्वारा बनाया गया हो?

Solution

  1. Step 1: Known probabilities define करें

    P(M₁) = 0.4, P(M₂) = 0.6
    P(D|M₁) = 0.02, P(D|M₂) = 0.03

  2. Step 2: Bayes’ Theorem apply करें

    P(M₁|D) = [P(D|M₁) × P(M₁)] / [P(D|M₁)P(M₁) + P(D|M₂)P(M₂)]

  3. Step 3: Substitute करें और simplify करें

    = (0.02 × 0.4) / [(0.02 × 0.4) + (0.03 × 0.6)]
    = 0.008 / (0.008 + 0.018) = 0.008 / 0.026 = 0.3077

  4. Final Answer:

    30.77% (≈ 0.31)

  5. Quick Check:

    M₁ total defective bulbs में 0.008 contribute करता है, total defective rate 0.026 है → 0.008/0.026 = 0.3077 ✅

Quick Variations

1. Medical Test: Positive test मिलने पर disease होने की probability निकालना।

2. Quality Control: Defective product किस machine ने बनाया, इसकी probability।

3. Email Classification: किसी email में specific words हों तो उसके spam होने की probability।

Trick to Always Use

  • Step 1: सभी hypotheses (A, A′) और उनकी priors पहचानें।
  • Step 2: P(B|A) और P(B|A′) निकालें - यानी evidence की likelihoods।
  • Step 3: Bayes’ formula apply करें और numerator को total evidence probability से divide करें।
  • Step 4: Result को आसानी से समझने के लिए percentage में convert करें।

Summary

Bayes’ Theorem pattern में:

  • यह reverse probabilities जैसे P(A|B) निकालने में मदद करता है।
  • Formula: P(A|B) = [P(B|A) × P(A)] / [P(B|A)P(A) + P(B|A′)P(A′)].
  • Priors (P(A)), likelihoods (P(B|A)) और total evidence probability को सही identify करें।
  • New evidence मिलने पर belief को systematically update करने में उपयोगी।
  • Quick Check: Denominator हमेशा evidence की total probability के बराबर होना चाहिए।

Practice

(1/5)
1. Box A contains 3 red and 2 blue balls. Box B contains 1 red and 4 blue balls. A box is chosen at random and one ball drawn; it is red. What is the probability the ball came from Box A?
easy
A. 3/4
B. 1/4
C. 1/2
D. 2/3

Solution

  1. Step 1: Write priors and likelihoods

    P(A)=1/2, P(B)=1/2. P(red|A)=3/5, P(red|B)=1/5.

  2. Step 2: Apply Bayes' rule

    P(A|red) = [P(red|A)P(A)] / [P(red|A)P(A) + P(red|B)P(B)].

  3. Step 3: Substitute

    =(3/5×1/2) / [(3/5×1/2) + (1/5×1/2)] = (3/10) / (4/10) = 3/4.

  4. Final Answer:

    3/4 → Option A.

  5. Quick Check:

    Box A is more likely to produce red (3/5 vs 1/5), so posterior > 1/2 → 3/4 ✅
Hint: Posterior ∝ prior × likelihood; normalize across boxes.
Common Mistakes: Forgetting to weight by the prior probability of choosing each box.
2. A disease has prevalence 1%. A test has 99% sensitivity (P(+|disease)=0.99) and 5% false positive rate (P(+|no disease)=0.05). If a randomly chosen person tests positive, what is the probability they actually have the disease?
easy
A. 1/20
B. 1/6
C. 1/10
D. 1/4

Solution

  1. Step 1: Write priors and likelihoods

    P(D)=0.01, P(not D)=0.99. P(+|D)=0.99, P(+|not D)=0.05.

  2. Step 2: Bayes' formula

    P(D|+) = P(+|D)P(D) / [P(+|D)P(D) + P(+|not D)P(not D)].

  3. Step 3: Substitute and compute

    = (0.99×0.01) / (0.99×0.01 + 0.05×0.99) = 0.0099 / (0.0099 + 0.0495) = 0.0099 / 0.0594 = 1/6 (≈0.1667).

  4. Final Answer:

    1/6 → Option B.

  5. Quick Check:

    Low prevalence makes false positives dominate despite high sensitivity → posterior ~16.7% ✅
Hint: Compute numerator = sensitivity×prevalence; denominator = numerator + falsePositiveRate×(1-prevalence).
Common Mistakes: Ignoring prevalence and treating sensitivity as posterior directly.
3. Machine M1 produces 40% of items with 2% defective rate; M2 produces 60% with 3% defective rate. A randomly chosen item is defective. What is the probability it came from M1?
easy
A. 0.5
B. 0.25
C. 0.3077
D. 0.6923

Solution

  1. Step 1: Write priors and likelihoods

    P(M1)=0.40, P(M2)=0.60. P(D|M1)=0.02, P(D|M2)=0.03.

  2. Step 2: Bayes' formula

    P(M1|D) = P(D|M1)P(M1) / [P(D|M1)P(M1) + P(D|M2)P(M2)].

  3. Step 3: Substitute

    =(0.02×0.40) / (0.02×0.40 + 0.03×0.60) = 0.008 / (0.008 + 0.018) = 0.008 / 0.026 = 0.3077.

  4. Final Answer:

    0.3077 → Option C.

  5. Quick Check:

    M1 contributes 0.008 of defects vs total 0.026 → 0.008/0.026 ≈ 30.77% ✅
Hint: Posterior = (likelihood×prior) / sum(likelihood×prior) across sources.
Common Mistakes: Using raw defect rates without weighting by production share.
4. Urn U1 has 2 white and 3 black balls; urn U2 has 4 white and 1 black. An urn is chosen with P(U1)=1/3 and P(U2)=2/3. A drawn ball is white. What is the probability it came from U1?
medium
A. 1/5
B. 2/5
C. 3/5
D. 4/5

Solution

  1. Step 1: Priors and likelihoods

    P(U1)=1/3, P(U2)=2/3. P(white|U1)=2/5, P(white|U2)=4/5.

  2. Step 2: Bayes' rule

    P(U1|white) = [P(white|U1)P(U1)] / [P(white|U1)P(U1) + P(white|U2)P(U2)].

  3. Step 3: Compute

    =(2/5×1/3) / [(2/5×1/3) + (4/5×2/3)] = (2/15) / (2/15 + 8/15) = (2/15) / (10/15) = 1/5.

  4. Final Answer:

    1/5 → Option A.

  5. Quick Check:

    U2 is more likely a priori and also likelier to give white, so posterior for U1 is small → 1/5 ✅
Hint: Compute contributions to evidence from each urn, then take ratio for the target urn.
Common Mistakes: Failing to multiply likelihood by the correct prior for each urn.
5. Three machines M1, M2, M3 produce 20%, 30%, 50% of items with defect rates 1%, 2%, 4% respectively. If an item is defective, what is the probability it came from M3?
medium
A. 1/3
B. 5/14
C. 2/7
D. 5/7

Solution

  1. Step 1: Compute contributions

    Numerators: M1 → 0.01×0.20=0.002; M2 → 0.02×0.30=0.006; M3 → 0.04×0.50=0.020.

  2. Step 2: Total defective probability

    Sum = 0.002 + 0.006 + 0.020 = 0.028.

  3. Step 3: Bayes' posterior for M3

    P(M3|defect) = 0.020 / 0.028 = 20/28 = 5/7 ≈ 0.7143.

  4. Final Answer:

    5/7 → Option D.

  5. Quick Check:

    M3 contributes the largest share of defects (0.02 of 0.028) → ~71.43% ✅
Hint: For multiple sources, compute likelihood×prior for each, then normalize by their sum.
Common Mistakes: Neglecting any machine's contribution when calculating the total defective probability.