Step-by-Step Solution

Topic: Total Probability & Bayes’ Theorem

1. Define Events & Probabilities

Events:

• • E₁: Airplane Crash
• • E₂: No Crash
• • A: Passengers survive

Probabilities (Convert % to Decimals):

P(E₁) = 0.00001% = 0.00001 / 100 = 0.0000001

P(E₂) = 1 – P(E₁) = 1 – 0.0000001 = 0.9999999

Conditional Probabilities:

P(A | E₁) = 95% = 0.95 (Survivors if crash)

P(A | E₂) = 1 (All survive if no crash)

(I) PROBABILITY OF NO CRASH

We need P(E₂).

P(E₂) = 1 – P(Crash)

P(E₂) = 1 – 0.0000001

(II) SUM OF CONDITIONAL PROBABILITIES

Calculate P(A | E₁) + P(A | E₂)

= 0.95 + 1

(III)(A) TOTAL PROBABILITY OF SURVIVAL P(A)

Using the Law of Total Probability:

Substitute the values:

P(A) = (0.0000001 × 0.95) + (0.9999999 × 1)

P(A) = 0.000000095 + 0.9999999

(III)(B) PROBABILITY OF NO CRASH (Given Survival)

We need P(E₂ | A) using Bayes’ Theorem:

Numerator: 0.9999999 × 1 = 0.9999999

Denominator: 0.999999995 (from Part III A)

P(E₂ | A) = 0.9999999 / 0.999999995

⚠️ Silly Mistake Audit: Percentage Conversion

Watch the decimal point!
The most common error here is treating 0.00001% as 0.00001.
Remember:
0.00001% = 0.00001 / 100 = 0.0000001.
Using the wrong value changes P(A) significantly!