Calculate the conditional probability of a die showing six given the man's truthful reporting | Step-by-Step Solution

🎲 Let's Solve This Bayes' Theorem Problem Together!

1. What We're Solving:

We need to find the probability that a die actually shows a six, given that a man (who tells the truth 3 out of 4 times) reports seeing a six.

2. The Approach:

This is a classic Bayes' Theorem problem! We're dealing with conditional probability - we want to "reverse" our thinking. Instead of asking "If the die shows six, what's the chance he reports it correctly?" we're asking "If he reports six, what's the chance the die actually shows six?"

The man could report "six" in TWO scenarios:

• The die IS a six AND he tells the truth
• The die is NOT a six AND he lies

3. Step-by-Step Solution:

Step 1: Define our events

• Let A = "Die actually shows a six"
• Let B = "Man reports a six"

We want to find P(A|B) - the probability the die shows six GIVEN that he reports six.

Step 2: Gather our known information

• P(man tells truth) = 3/4, so P(man lies) = 1/4
• P(die shows six) = 1/6, so P(die doesn't show six) = 5/6

Step 3: Find P(B) - the probability he reports "six" He can report "six" in two ways:

• Case 1: Die shows six AND he tells truth = (1/6) × (3/4) = 3/24
• Case 2: Die shows something else AND he lies = (5/6) × (1/4) = 5/24

So P(B) = 3/24 + 5/24 = 8/24 = 1/3

Step 4: Apply Bayes' Theorem P(A|B) = P(B|A) × P(A) / P(B)

Where:

• P(B|A) = probability he reports six when die shows six = 3/4 (he tells truth)
• P(A) = 1/6 (die shows six)
• P(B) = 1/3 (calculated above)

P(A|B) = (3/4) × (1/6) ÷ (1/3) = (3/24) ÷ (1/3) = (3/24) × 3 = 9/24 = 3/8

4. The Answer:

The probability that the die actually shows a six is 3/8 or 0.375 (37.5%)

Notice something interesting: Even though he's mostly truthful, the probability is less than 50%! This happens because it's much more likely for the die to show something other than six (5 out of 6 times).

5. Memory Tip:

🧠 Remember: In Bayes' problems, always consider ALL the ways the observed event can happen! The man reporting "six" doesn't just depend on his honesty - it also depends on what the die actually showed. Sometimes the "rare event" (rolling a six) makes the final probability lower than you'd expect!

Great job working through this challenging concept! Bayes' Theorem takes practice, but you're building really important statistical reasoning skills! 🌟