Financial Risk Manager Part 1

Explanation

This is a binomial probability problem where:

• n = 9 (number of trials/individuals)
• p = 0.6 (probability of survival)
• q = 0.4 (probability of not surviving)
• X = number of survivors

We need to find P(X ≥ 8), which means at least 8 survivors out of 9.

Step 1: Break down the probability

P(X ≥ 8) = P(X = 8) + P(X = 9)

Step 2: Calculate P(X = 8)

Using the binomial formula: $P(X = 8) = \binom{9}{8} \cdot (0.6)^8 \cdot (0.4)^1$ $P(X = 8) = 9 \cdot (0.6)^8 \cdot (0.4)$

Step 3: Calculate P(X = 9)

$P(X = 9) = \binom{9}{9} \cdot (0.6)^9 \cdot (0.4)^0$ $P(X = 9) = 1 \cdot (0.6)^9 \cdot 1$

Step 4: Numerical calculation

• (0.6)^8 = 0.01679616
• (0.6)^9 = 0.010077696

P(X = 8) = 9 × 0.01679616 × 0.4 = 9 × 0.006718464 = 0.060466176

P(X = 9) = 0.010077696

P(X ≥ 8) = 0.060466176 + 0.010077696 = 0.070543872 ≈ 0.071

Step 5: Match with options

0.071 rounds to 0.07, which corresponds to option B.

Key points:

• This is a binomial distribution problem
• The events are independent (infection of one person doesn't affect others)
• The trials are identical (same survival probability for each person)
• The calculation uses the binomial probability formula: $P(X = x) = \binom{n}{x} p^x q^{n-x}$