People's Education Press High School Mathematics: Elective Compulsory Course 3 (A Edition): From Conditional Probability to Bayesian Decision-Making: Mathematical Tools for Causal Reasoning

Imagine you are a digital archaeologist. When you encounter a corrupted communication code (outcome $B$), your task is to infer the original instruction sent by the source (cause $A$). This logic of reasoning backward from 'effect' to 'cause' lies at the heart of how modern artificial intelligence handles uncertainty.

Starting from the definition of conditional probability $P(B|A)$, we can not only calculate the evolution of sequential events but also decompose global complexity into weighted sums of local conditions throughthe Law of Total Probabilityinto weighted sums of local conditions. AndBayes' Theoremis the crown of this theory—it allows us to continuously revise our prior knowledge (prior) based on new information (posterior), enabling dynamic cognitive evolution.

The Logical Three-Step Leap in Probability Theory

Step One: Local Dependence (Multiplication Rule)
When the occurrence of event $B$ depends on $A$, their joint probability is no longer a simple product, but $P(AB) = P(A)P(B|A)$. This is especially critical in sampling without replacement.

Step Two: Structural Decomposition (Law of Total Probability)
For complex macro-level events $B$, we project them onto different backgrounds $A_i$. The Law of Total Probability $P(B) = \sum P(A_i)P(B|A_i)$ tells us: the overall probability equals the expected value of local conditional probabilities.

Step Three: Causal Reverse Inference (Bayes' Theorem)
This is the formula of wisdom. It revises the 'prior probability' $P(A_i)$ (experience before the experiment) through the 'likelihood' $P(B|A_i)$ into the 'posterior probability' $P(A_i|B)$.

The Law of Total Probability is predictive—reasoning forward from cause to effect—while Bayes' Theorem is decision-making—reasoning backward from effect to cause. Together, they form the mathematical foundation of modern risk management and medical diagnosis.

$$P(A_i | B) = \frac{P(A_i)P(B | A_i)}{\sum_{k=1}^n P(A_k)P(B | A_k)}$$

QUESTION 1

Let $A \subseteq B$, with $P(A) = 0.3$ and $P(B) = 0.6$. Find the values of $P(B|A)$ and $P(A|B)$.

$P(B|A) = 1, P(A|B) = 0.5$

$P(B|A) = 0.5, P(A|B) = 1$

$P(B|A) = 0.3, P(A|B) = 0.6$

$P(B|A) = 1, P(A|B) = 1$

QUESTION 2

Draw two cards consecutively without replacement from a standard 52-card deck (no jokers). Given that the first card drawn is an Ace, what is the probability that the second card is also an Ace?

$3/51$

$4/52$

$3/52$

$4/51$

QUESTION 3

A bag contains 7 white and 3 black balls. Draw two balls without replacement. Given that the first ball drawn is white, what is the probability that the second ball is also white?

$2/3$

$7/10$

$6/9$

$7/9$

QUESTION 4

Zhang Jun has insight into 9 out of 12 questions (probability of getting it right is 0.9), and guesses blindly on the other 3 (probability of guessing correctly is 0.25). What is the probability of getting a randomly selected question right?

$0.7375$

$0.85$

$0.575$

$0.9125$

QUESTION 5

Two batches of products: the first accounts for 40% (defect rate 5%), the second 60% (defect rate 4%). What is the probability of selecting a non-defective item at random from the mixed batch?

$0.956$

$0.044$

$0.96$

$0.95$

QUESTION 6

Building on the previous question, given that a non-defective item was selected, what is the probability it came from the first batch?

$0.38/0.956$

$0.4/0.956$

$0.02/0.956$

$0.5/0.956$

QUESTION 7

In regions A, B, and C, flu rates are 6%, 5%, and 4% respectively, with population ratios of 5:7:8. What is the probability that a randomly selected person has the flu?

$4.85\%$

$5.0\%$

$15\%$

$5.25\%$

QUESTION 8

A marksman's distribution for hitting 6–10 rings is $P$: 0.05, 0.15, 0.25, 0.35, 0.20. What is the probability of achieving an excellent score (hitting 9 or 10 rings)?

$0.55$

$0.35$

$0.20$

$0.80$

QUESTION 9

A discrete random variable $X$ has distribution: $P(X=0) = 0.36$, $P(X=1) = 1 - 2q$, $P(X=2) = q^2$. Find the value of constant $q$.

$0.2$

$0.8$

$1.8$

$0.32$