In-depth Notes on Conditional Probability and Related Topics

1.3 Conditional Probability

• Definition: Conditional probability adjusts the probability model to reflect partial information about an experimental outcome.
• The sample space of this corrected model is smaller than the original model's.

Concepts:

• Prior Probability: The probability of event A occurring given the background of all events.
  • Notation: $P(A|S) = P(A)$
• Posterior Probability: The probability of event A occurring given the new event B.
  • Notation: $P(A|B)$

Example 1.9:

• Consider testing two Integrated Circuits (IC) from the same silicon wafer with outcomes accepted (a) or rejected (r).

• The sample space S = {rr, ra, ar, aa}.

  • Let B be the event the first chip is rejected: $B = {rr, ra}$
  • Let A be the event the second chip fails: $A = {rr, ar}$

• These chips are from a high-quality production line, thus prior probability $P[A]$ is very low.

• However, if some silicon wafers are contaminated with dust, the failure of chips increases.

• Given event B occurred, the probability of event A occurring is higher than prior probability $P[A]$ because it’s likely that dust contaminated the entire wafer.

Definition 1.5: Conditional Probability

• $P(A|B) = \frac{P(A \cap B)}{P(B)}$
  • This represents the probability of event A given B.

Example Calculations:

1. Given the sample space S = {x| 0<x<= 10} with integers {1, 2, …, 10}:
   • Let A = {1, 2, 3} and B = {2, 4, 6, 8, 10}.
   • Probability calculations:
     • $P(B) = \frac{5}{10} = 0.5$
     • $P(A) = \frac{5}{10} = 0.5$
     • $P(A \cap B) = \frac{2}{10} = 0.2$
   • Therefore conditional probability:
     • $P(A|B) = \frac{0.2}{0.5} = 0.4$

Properties of Conditional Probability:

1. $P[A|B] \geq 0$
2. $P[B|B] = 1$
3. If P(A) > 0 and P(B) > 0, then:
   • $P(A|B) \cdot P(B) = P(A \cap B)$

Example 1.10:

• Given prior probabilities: $P[rr] = 0.01, P[ra] = 0.01, P[ar] = 0.01, P[aa] = 0.97$.
• Request to find probabilities of A and B and their conditional probability.

Example 1.12 and Example 1.13:

• Testing four coins for heads (h) or tails (t) generates a sample space with 16 outcomes.
• Each event can be analyzed separately, which constitutes a partition of the sample space.

1.4 Partition and Law of Total Probability:

• A partition splits the sample space into mutually exclusive sets.
• The law of total probability states the probability of an event A can be expressed as the sum of the probabilities of A over the individual components of a partition.

Example 1.15:

• Classifying emails into categories based on length and type yields a sample space S = {lt, bt, li, bi, lv, bv}.
• Example shows how to apply the law of total probability to find probabilities based on these classifications.

Bayes’ Theorem (Theorem 1.10):

• $P[B|A] = \frac{P[A|B] \cdot P[B]}{P[A]}$

Example 1.17:

• Calculating probabilities for acceptable resistors from different manufacturing machines considering their production rates and quality qualifications.

Independence of Events:

• Two events are independent if the occurrence of one does not affect the probability of the other.

• Examples demonstrate independence (or lack thereof) across several scenarios.