It is known that $20 \%$ of the students in a school report being left-handed. Students at this school are selected at random and asked about their handedness. Find the probability that the first left-handed student encountered occurs on the sixth student selected.
$P(X=6)=[?]$
Round to the nearest thousandth.
Answer (2)
Recognize the problem as a geometric distribution.
Apply the geometric distribution formula: P ( X = k ) = ( 1 − p ) k − 1 ∗ p .
Substitute p = 0.2 and k = 6 into the formula: P ( X = 6 ) = ( 0.8 ) 5 ∗ 0.2 .
Calculate the probability and round to the nearest thousandth: 0.066 .
Explanation
Understand the problem and provided data We are given that 20% of students are left-handed, meaning the probability of a student being left-handed is p = 0.2 . We want to find the probability that the first left-handed student is the sixth student we encounter. This means the first five students are not left-handed, and the sixth student is left-handed.
Identify the distribution and formula This is a geometric distribution problem. The probability mass function for a geometric distribution is given by: P ( X = k ) = ( 1 − p ) k − 1 × p where p is the probability of success (finding a left-handed student) and k is the number of trials until the first success.
Apply the formula In this case, p = 0.2 and k = 6 . Therefore, we want to calculate P ( X = 6 ) :
P ( X = 6 ) = ( 1 − 0.2 ) 6 − 1 × 0.2 = ( 0.8 ) 5 × 0.2
Calculate the probability Calculating this value: ( 0.8 ) 5 × 0.2 = 0.32768 × 0.2 = 0.065536
Round to the nearest thousandth Rounding the result to the nearest thousandth, we get: P ( X = 6 ) ≈ 0.066
Examples
Imagine you're trying to recruit members for a club, and you know that 20% of students are interested in joining. You randomly approach students and ask if they're interested. The probability that the first student who says 'yes' is the sixth student you ask is about 0.066. This kind of calculation helps in planning recruitment strategies or estimating how many people you might need to approach before finding someone interested.
The probability that the first left-handed student encountered occurs on the sixth student selected is approximately 0.066. This is calculated using the geometric distribution formula, with a success probability of 0.2 and requiring five failures followed by a success. Hence, P ( X = 6 ) = 0.066 .
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