Probability: Conditional Probability & Bayes' Theorem

TAKEAWAYS:

• Conditional probability helps find the probability of an event given another event has already occurred.
• Bayes' Theorem connects prior probabilities, conditional probabilities, and posterior probabilities.
• Tree diagrams and tables simplify complex probability problems.
• Real-world applications include medical testing, spam filtering, and risk assessment.
• Always check if events are independent before applying formulas.
• Practice NEB-style questions to recognize when to use conditional probability vs. Bayes' Theorem.

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1. Introduction to Conditional Probability

Probability is the chance of an event happening. Sometimes, we want to find the probability of an event given that another event has already occurred. This is called conditional probability.

Definition

The probability of event A happening given that event B has already happened is written as: $$P(A|B)=\frac{P(A\cap B)}{P(B)}$$ where:

• $P(A|B)$ = Probability of A given B
• $P(A\cap B)$ = Probability of both A and B happening
• $P(B)$ = Probability of B happening

Example 1: Basic Conditional Probability

A bag contains 5 red balls and 3 blue balls. Two balls are drawn without replacement. What is the probability that the second ball is red given that the first ball drawn was blue?

Solution:

1. Total balls = 5 red + 3 blue = 8 balls.
2. Probability of first ball being blue: $$P(B)=\frac{3}{8}$$
3. If the first ball is blue, remaining balls = 5 red + 2 blue = 7 balls.
4. Probability of second ball being red given first was blue: $$P(R|B)=\frac{5}{7}$$

Answer: $\frac{5}{7}$

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2. Visualizing Conditional Probability

This pie chart shows:

• $P(B)=\frac{3}{8}$
• $P(A\cap B)=\frac{1}{8}$
• $P(A|B)=\frac{P(A\cap B)}{P(B)}=\frac{1/8}{3/8}=\frac{1}{3}$

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3. Independent vs. Dependent Events

• Independent Events: The occurrence of one event does not affect the other. Example: Rolling a die and flipping a coin. $$P(A|B)=P(A)$$

• Dependent Events: The occurrence of one event affects the other. Example: Drawing balls without replacement.

Comparison Table:

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4. Bayes' Theorem

Bayes' Theorem helps update probabilities based on new information. It is widely used in medical testing, spam detection, and machine learning.

Formula:

$$P(A|B)=\frac{P(B|A)\times P(A)}{P(B)}$$ where:

• $P(A|B)$ = Posterior probability (what we want to find)
• $P(B|A)$ = Likelihood (probability of B given A)
• $P(A)$ = Prior probability (initial probability of A)
• $P(B)$ = Marginal probability (total probability of B)

Example 2: Medical Testing

A disease affects 1% of the population. A test for the disease is 95% accurate (95% true positive rate and 95% true negative rate). If a person tests positive, what is the probability they actually have the disease?

Solution:

1. Let:

   • $D$ = Person has the disease
   • $\neg D$ = Person does not have the disease
   • $T^{+}$ = Test is positive

2. Given:

   • $P(D)=0.01$
   • $P(\neg D)=0.99$
   • $P(T^{+}|D)=0.95$ (True positive rate)
   • $P(T^{-}|\neg D)=0.95$ (True negative rate), so $P(T^{+}|\neg D)=0.05$

3. Find $P(D|T^{+})$: $$P(D|T^{+})=\frac{P(T^{+}|D)\times P(D)}{P(T^{+})}$$ where $P(T^{+})=P(T^{+}|D)\times P(D)+P(T^{+}|\neg D)\times P(\neg D)$.

4. Calculate: $$P(T^{+})=(0.95\times 0.01)+(0.05\times 0.99)=0.0095+0.0495=0.059$$ $$P(D|T^{+})=\frac{0.95\times 0.01}{0.059}=\frac{0.0095}{0.059}\approx 0.161\text{ or }16.1\%$$

Answer: Only 16.1% of people who test positive actually have the disease!

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5. Tree Diagrams for Probability

Tree diagrams help visualize conditional probabilities step-by-step.

Example 3: Two-Stage Probability

A factory produces 10% defective items. A quality check catches 90% of defective items but also mislabels 5% of good items as defective. What is the probability that an item is actually defective given it was flagged by the check?

Solution:

1. Probability of flagged item being defective: $$P(D|F)=\frac{P(F|D)\times P(D)}{P(F)}$$ where $P(F)=P(F|D)\times P(D)+P(F|\neg D)\times P(\neg D)$.

2. Calculate: $$P(F)=(0.9\times 0.1)+(0.05\times 0.9)=0.09+0.045=0.135$$ $$P(D|F)=\frac{0.9\times 0.1}{0.135}=\frac{0.09}{0.135}\approx 0.667\text{ or }66.7\%$$

Answer: 66.7% of flagged items are actually defective.

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6. Applications of Conditional Probability and Bayes' Theorem

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7. Common Mistakes to Avoid

1. Ignoring Independence: Assume events are independent when they are not.
2. Misapplying Bayes' Theorem: Forgetting to calculate the total probability $P(B)$.
3. Confusing $P(A|B)$ and $P(B|A)$: These are not the same!
4. Incorrect Sample Space: Not updating the sample space after an event occurs (e.g., drawing without replacement).

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8. NEB-Style Questions

Short Answer Questions

1. Define conditional probability with an example.
2. What is the difference between independent and dependent events?
3. State Bayes' Theorem and explain its components.

Long Answer Questions

1. A bag contains 4 red and 6 blue balls. Two balls are drawn with replacement. Find the probability that:

   • Both are red.
   • The second ball is blue given the first was red.
   • The first ball is red given both are blue.

2. In a class of 50 students, 20 study Maths, 15 study Physics, and 10 study both. If a student is selected at random:

   • What is the probability they study Maths given they study Physics?
   • What is the probability they study Physics given they study Maths?

3. A factory produces 5% defective items. A test detects 98% of defective items but also mislabels 2% of good items as defective. If a randomly selected item tests positive, what is the probability it is actually defective?

Objective Questions (MCQs)

1. If $P(A)=0.4$, $P(B)=0.5$, and $P(A\cap B)=0.2$, then $P(A|B)=$ ?

   • (A) 0.4
   • (B) 0.5
   • (C) 0.8
   • (D) 0.2

2. Bayes' Theorem is used to:

   • (A) Find marginal probability
   • (B) Update probabilities with new information
   • (C) Calculate independent events
   • (D) Find sample space

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Exam Tip

1. Always check if events are independent before applying formulas.
2. Draw tree diagrams or tables for complex problems to avoid confusion.
3. Memorize Bayes' Theorem formula but understand each component:
   • Prior probability ($P(A)$)
   • Likelihood ($P(B|A)$)
   • Marginal probability ($P(B)$)
4. Practice NEB-style questions to recognize when to use conditional probability vs. Bayes' Theorem.
5. Show all steps in your answer, even if the question seems simple. Partial credit is given for correct reasoning.

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Good luck with your NEB exam preparation! 🚀