Posts in category: Probability

Probability is a powerful tool for making informed decisions, especially when not everything is known. But what happens when you’re given partial information — like a positive test result, or getting a quiz question correct? Conditional probability helps us update our beliefs in such situations.

In this article, we’ll explore:

• What conditional probability really means
• Key rules: Chain Rule, Law of Total Probability, and Bayes’ Theorem
• 5 fully worked-out real-world examples

🧠 What is Conditional Probability?

Conditional probability is the probability of an event occurring given that another event has already occurred.

This means: out of the scenarios where B happens, how many also include A?

Example:
What is the chance that it’s an Ace given the card is red?

🔗 Chain Rule of Probability

The chain rule helps us build up joint probabilities using conditional ones:

P(A ∩ B) = P(A | B) × P(B)

P(A ∩ B ∩ C) = P(C | A ∩ B) × P(B | A) × P(A)

📊 Law of Total Probability

Used when outcomes arise from multiple mutually exclusive causes:

P(B) = P(B | A₁) × P(A₁) + P(B | A₂) × P(A₂) + ...

🔁 Bayes’ Theorem

Used to reverse conditional probabilities:

It shines in diagnosis, decision-making, and machine learning.

💡 Worked Example 1: Multiple Choice Theory

A student answers a multiple-choice question.

• Knows the concept: P(K) = 3/4
• Guessing correctly: P(C | ¬K) = 1/4
• Gets it right even if they know: P(C | K) = 9/10

What is P(K | C) — the chance they knew it given they got it correct?

Step 1: Use Bayes’ Theorem

P(C) = (9/10)(3/4) + (1/4)(1/4)
     = 27/40 + 1/16
     = (108 + 5) / 160 = 113 / 160

P(K | C) = (9/10 × 3/4) / (113/160)
         = (27/40) ÷ (113/160)
         = (27 × 160) / (40 × 113)
         = 54 / 59

🧪 Example 2: Medical Test Accuracy

A disease affects 1 in 1000. Test sensitivity = 99%, specificity = 98%. You test positive.

P(D) = 0.001,  P(¬D) = 0.999
P(T⁺ | D) = 0.99,  P(T⁺ | ¬D) = 0.02

P(T⁺) = 0.00099 + 0.01998 = 0.02097
P(D | T⁺) = 0.00099 / 0.02097 ≈ 0.0472

☔ Example 3: Rain and Umbrella

Friend carries an umbrella. What’s the chance it’s raining?

P(R) = 0.3,  P(U | R) = 0.9,  P(U | ¬R) = 0.2

P(U) = 0.27 + 0.14 = 0.41
P(R | U) = 0.27 / 0.41 ≈ 0.6585

🃏 Example 4: Cards

Probability of Ace given the card is red?

P(Ace ∩ Red) = 2/52,  P(Red) = 26/52
P(Ace | Red) = (2/52) / (26/52) = 2/26 = 1/13

🏭 Example 5: Faulty Factory Machine

• Machine A: 30%, defect = 2%
• Machine B: 50%, defect = 1%
• Machine C: 20%, defect = 3%

Find P(C | Defect)

P(D) = 0.006 + 0.005 + 0.006 = 0.017
P(C | D) = 0.006 / 0.017 ≈ 0.3529

🧠 Final Thoughts

Conditional probability helps us make better decisions with limited info.

• Chain Rule = builds from conditional steps
• Law of Total Probability = combines causes
• Bayes’ Theorem = reverses conditionals

Whether you’re a student, analyst, or clinician — these tools are essential to your decision-making toolkit.

Ever wondered how many ways an event can occur? That’s where counting in probability comes in. It’s a powerful tool that helps you calculate possible outcomes before diving into probability formulas.

In this guide, we’ll break down the essentials, explain the product and sum rules, and walk through easy-to-follow examples.

What is Counting in Probability?

Counting is the foundation of probability. It helps you determine how many outcomes meet a particular condition. Once you can count favorable outcomes, calculating probability becomes easy.

Example 1: Counting Dice Outcomes

Suppose you roll a fair 6-sided die:

• Q: How many possible outcomes are there?
  A: 6 outcomes → {1, 2, 3, 4, 5, 6}
• Q: How many outcomes are greater than 4?
  A: 2 outcomes → {5, 6}

Tip: Define your condition, then count only the outcomes that satisfy it.

The Product Rule of Counting (Step Rule)

Definition:

If an experiment has two independent steps:

• Step 1 has m outcomes
• Step 2 has n outcomes

Total outcomes = m × n

Example 2: Choosing Outfits

You have:

• 3 shirts (Red, Blue, Green)
• 2 pants (Black, White)

Q: How many outfit combinations can you make?
A: 3 × 2 = 6 combinations

Combinations: Red-Black, Red-White, Blue-Black, Blue-White, Green-Black, Green-White

Sum Rule of Counting

Definition:

If an outcome can come from either of two mutually exclusive events:

Total outcomes = m + n

Example 3: Choosing Snacks

A vending machine offers:

• 4 chips
• 3 chocolate bars

Q: How many different snacks can you choose if you pick either one chip or one bar?
A: 4 + 3 = 7 snacks

How Many Valid Bit Strings?

Problem:

A 6-bit string is valid if it:

• Starts with 01 OR
• Ends with 10

Solution Step-by-Step:

1. Total 6-bit strings: 2⁶ = 64
2. Start with “01”: Remaining 4 bits → 2⁴ = 16
3. End with “10”: First 4 bits → 2⁴ = 16
4. Both start with “01” and end with “10”: Middle 2 bits → 2² = 4

Valid strings = 16 + 16 – 4 = 28

Inclusion-Exclusion Rule

Definition:

To avoid double-counting overlapping sets:

|A ∪ B| = |A| + |B| – |A ∩ B|

Example 4: Tea and Coffee Lovers

• 10 people like tea
• 8 like coffee
• 3 like both

Q: How many people like either tea or coffee?
A: 10 + 8 – 3 = 15

Quick Recap: Core Counting Rules

Conclusion

Counting isn’t just math — it’s a practical tool for navigating uncertainty. From games to networks, knowing how to count helps you understand what’s likely — and what’s not.

Up Next: We’ll explore permutations and combinations — the next step in mastering probability.
Permutations and Combinations

Got Questions?

Leave a comment below — I’d love to hear your thoughts or explain anything you’d like in more detail!