Question

of all the fish in a certain river, 20 percent are salmon. once a year, people can purchase a fishing license that allows them to catch up to 8 fish. assume each catch is independent. which of the following represents the probability of needing to catch 8 fish to get the first salmon?
a 0.2
b \\(\frac{1}{0.2}\\)
c \\(0.2^8\\)
d \\(0.2(0.8)^7\\)
e \\(0.8(0.2)^7\\)

Explanation:

Step1: Identify Distribution Type

This is a geometric distribution problem, where we want the probability that the first success (catching a salmon) occurs on the 8th trial (catch). The probability of success (salmon) is \( p = 0.2 \), and failure (not salmon) is \( q = 1 - p = 0.8 \).

Step2: Geometric Probability Formula

The formula for the probability that the first success occurs on the \( n \)-th trial is \( P(X = n) = q^{n - 1}p \). Here, \( n = 8 \), so \( P(X = 8) = (0.8)^{8 - 1}(0.2) = 0.8^{7}(0.2) \)? Wait, no—wait, wait. Wait, no: Wait, the first success on the 8th trial means the first 7 are failures, then the 8th is success. So failure probability is \( 0.8 \), success is \( 0.2 \). So the probability is \( (0.8)^7 \times 0.2 \)? Wait, no, wait the options: Let's check the options. Option D is \( 0.2(0.8)^7 \), which is the same as \( (0.8)^7 \times 0.2 \). Wait, but let's re-express: The geometric distribution formula is \( P(X = k) = (1 - p)^{k - 1}p \), where \( k \) is the number of trials until first success. So here, \( k = 8 \), \( p = 0.2 \), so \( (1 - 0.2)^{8 - 1} \times 0.2 = 0.8^7 \times 0.2 = 0.2(0.8)^7 \), which is option D. Wait, but let's check the options again. Wait, option D is \( 0.2(0.8)^7 \), which matches. Wait, but let's confirm: The first 7 catches are not salmon (probability 0.8 each), and the 8th is salmon (probability 0.2). So the probability is \( (0.8)^7 \times 0.2 = 0.2(0.8)^7 \), which is option D.

Answer:

D. \( 0.2(0.8)^7 \)