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probability.txt (2503B)

      1 ========================================================================
      2 Probability
      3 ========================================================================
      4 
      5 Bayes Theoreom
      6 
      7 P(A∣B)= P(B∣A)⋅P(A) / P(B)​
      8 
      9 Where:
     10 
     11 P(A|B) = Posterior probability (probability of A given B has occurred)
     12 P(B|A) = Likelihood (probability of B given A is true)
     13 P(A) = Prior probability (initial belief about A before seeing evidence)
     14 P(B) = Evidence (total probability of observing B)
     15 
     16 Bayes' theorem lets you reverse conditional probabilities. If you know the probability of observing some evidence given a hypothesis, you can calculate the probability that the hypothesis is true given that you've observed the evidence.
     17 
     18 Practical Example:
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     20 Imagine you're testing for a rare disease:
     21 
     22 The disease affects 1% of the population: P(Disease) = 0.01
     23 The test correctly identifies the disease 99% of the time: P(Positive|Disease) = 0.99
     24 The test has a 5% false positive rate: P(Positive|No Disease) = 0.05
     25 
     26 If you test positive, what's the actual probability you have the disease?
     27 
     28 Using Bayes:
     29 
     30 𝑃(Disease|Positive) = 0.99 × 0.01 / (0.99 × 0.01) + (0.05 × 0.99)
     31 = 0.0099 / 0.0594 ≈ 16.7%
     32 
     33 Even with a positive test, there's only about a 17% chance you actually have the disease—because the disease is so rare that false positives are common relative to true positives.
     34 
     35 This is why Bayes' theorem is so powerful: it captures how evidence should change our confidence in a hypothesis, accounting for base rates and test accuracy.
     36 
     37 
     38 Question:
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     40 A machine has 4 independent components, each of which fails during a shift with probability 0.05. What is the probability that at least one fails?
     41 A: 5%  B: 18.5%  C: 19.0%  D: 20%  E: 81.5%
     42 
     43 
     44 Probability of at least 1 failing:
     45 
     46 0. so the component has a success probability of 1 - 0.05 = 0.95
     47 1. calculate the probability of all components functioning successfully:
     48 0.95 x 0.95 x 0.95 x 0.95 = 0.815
     49 2. so 1 - 0.815 is 0.185 is the chance of at least 1 component fails
     50 
     51 Probability of at least 2 failing:
     52 
     53 "At least 2 fails" is that same bundle with one case removed — exactly 1. So:
     54 0.05 x 0.95 x 0.95 x 0.95 = 0.0429
     55 But there are four different components that could be the one that failed, and those are four separate outcomes, so add them up:
     56 4 × 0.0429 = 0.1715
     57 Then subtract:
     58 0.185 − 0.1715 = 0.0140, about 1.4%
     59 
     60 Same logic extends upward: to get "at least 3", you'd take 0.0140 and subtract P(exactly 2). Each step peels off one more case from the bundle.