Financial Risk Manager Part 1

To calculate the probability of having exactly 3 sunny days in the next 7 days, we use the Binomial Distribution formula:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

Where:

• n = 7 (number of days)
• k = 3 (number of sunny days)
• p = 0.9 (probability of sunny day)

$P(3) = \binom{7}{3} \times 0.9^3 \times (1 - 0.9)^{7-3}$

$P(3) = \frac{7!}{3!(7-3)!} \times 0.9^3 \times 0.1^4$

$P(3) = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} \times 0.729 \times 0.0001$

$P(3) = 35 \times 0.729 \times 0.0001 = 0.00255$

This is a binomial probability problem where we're looking for exactly 3 successes (sunny days) in 7 independent trials (days), with each trial having a 90% probability of success.