Binomial Probability: Find the Chance 22 of 30 90% Germination Sunflower Seeds Grow

Question

Carol's Seed Haven claims that 90% of its sunflower seeds will germinate. Suppose that a company orders a case. Siti buys a packet with 30 sunflower seeds from Carol's Seed Haven and plants them in her garden. What is the probability that exactly 22 seeds will germinate?

Accepted Answer

0.0208

Answer

Step 1: Identify this as a binomial probability problem, since we have independent trials (each seed germinating is independent), a fixed number of trials (n=30), two outcomes (germinate or not), and a constant success probability (p=0.9, where success = seed germinates). Step 2: Recall the binomial probability formula: $P(X=k) = C(n,k) 	imes p^k 	imes (1-p)^{n-k}$, where $C(n,k)$ is the combination of n things taken k at a time. Step 3: Define the values: n=30, k=22, p=0.9, so 1-p=0.1. Step 4: Calculate the combination $C(30,22) = C(30,8) = \frac{30!}{22!8!} = 5852925$. Step 5: Compute $p^k = 0.9^{22} \approx 0.1060449937$, and $(1-p)^{n-k} = 0.1^{8} = 0.00000001$. Step 6: Multiply the three values together: $5852925 	imes 0.1060449937 	imes 0.00000001 \approx 0.0208$.