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Difficulty: MediumComplementary Probability and At-Least-One Scenarios

An automated security gate opens successfully on any given approach with a probability of 0.80.8, independently of other attempts. If a driver approaches the gate 33 times, the probability that the gate opens successfully at least once is 0.9920.992.

Answer: Answer

Answer

The statement is true because the probability of failing all 33 independent attempts is (1−0.8)3=0.008(1 - 0.8)^3 = 0.008, making the probability of at least one successful opening 1−0.008=0.9921 - 0.008 = 0.992.
The statement accurately reflects the complementary probability principle P(at least one)=1−P(none)P(\text{at least one}) = 1 - P(\text{none}). For 33 independent trials with single-trial failure probability 0.20.2, the probability of all trials failing is 0.23=0.0080.2^3 = 0.008. Subtracting this from 11 yields 0.9920.992.

Step-by-Step Solution

1
Find the probability of failure on a single trial
P(failure)=1−0.8=0.2P(\text{failure}) = 1 - 0.8 = 0.2
The failure of a trial is the complement of its success.
2
Calculate the probability of zero successes across 3 independent trials
P(0 successes)=(0.2)3=0.008P(\text{0 successes}) = (0.2)^3 = 0.008
For independent events, the joint probability is the product of individual probabilities.
3
Apply the complementary probability rule to find the probability of at least one success
P(at least 1 success)=1−0.008=0.992P(\text{at least 1 success}) = 1 - 0.008 = 0.992
The event 'at least one success' is the exact complement of 'zero successes'.

Key Concept

Complementary Probability and At-Least-One Scenarios
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Topics

Complementary Probability and At-Least-One Scenarios

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