There are 3 radar stations and the probability of a single radar station detecting an enemy plane is 0.65 and the probability of not detecting an enemy plane is 0.35. What is the probability that the number of stations that detect a plane is no more than 2? Round your answer to nearest hundredth.
A)0.73
B)0.44
C)0.65
D)0.16
E)0.15

Question

Quizplus · Accepted Answer

The probability of no more than 2 stations detecting a plane includes the scenarios where 0, 1, or 2 stations detect the plane. We calculate these probabilities using the binomial distribution formula, $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$, where $n$ is the number of trials (stations), $k$ is the number of successes (detections), and $p$ is the probability of success (detection).1. Probability of 0 stations detecting the plane: $P(X=0) = \binom{3}{0}(0.65)^0(0.35)^3 = 1 \times 1 \times 0.35^3 = 0.042875$.2. Probability of 1 station detecting the plane: $P(X=1) = \binom{3}{1}(0.65)^1(0.35)^2 = 3 \times 0.65 \times 0.35^2 = 0.227025$.3. Probability of 2 stations detecting the plane: $P(X=2) = \binom{3}{2}(0.65)^2(0.35)^1 = 3 \times 0.65^2 \times 0.35 = 0.439275$.Adding these probabilities gives the total probability of no more than 2 stations detecting the plane: $0.042875 + 0.227025 + 0.439275 = 0.709175$, which rounds to $0.71$, closest to option A, $0.73$.

There Are 3 Radar Stations and the Probability of a Single