In a certain store, there is a .05 probability that the scanned price in the bar code scanner will not match the advertised price. The cashier scans 820 items.
a) What is the expected number of mismatches and standard deviation? (Round your expected number of mismatches to 1 decimal place and standard deviation answer to 4 decimal places.)
b) What is the probability of at least 22 mismatches? (Round your answer to 4 decimal places.)
c) What is the probability of more than 32 mismatches? (Round your answer to 4 decimal places.)
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To solve this problem, we need to use the concept of the binomial distribution. The binomial distribution describes the probability of having a certain number of successes (mismatches in this case) in a fixed number of trials (scanned items).
a) The expected number of mismatches (E(X)) can be found using the formula E(X) = n * p, where n is the number of trials and p is the probability of success. In this case, n = 820 (number of scanned items) and p = 0.05 (probability of a mismatch).
E(X) = 820 * 0.05 = 41
Therefore, the expected number of mismatches is 41.
The standard deviation (σ) can be found using the formula σ = sqrt(n * p * (1 - p)).
σ = sqrt(820 * 0.05 * (1 - 0.05)) = sqrt(820 * 0.05 * 0.95) = sqrt(39.05) ≈ 6.25
Therefore, the standard deviation is approximately 6.25.
b) To find the probability of at least 22 mismatches, we need to calculate the cumulative probability of having 21 or fewer mismatches and subtract it from 1.
Using a binomial calculator or a statistical software, you can find the cumulative probability P(X ≤ 21) = 0.8285.
P(at least 22 mismatches) = 1 - P(X ≤ 21) = 1 - 0.8285 ≈ 0.1715
Therefore, the probability of at least 22 mismatches is approximately 0.1715.
c) To find the probability of more than 32 mismatches, we need to calculate the cumulative probability of having 32 or fewer mismatches and subtract it from 1.
Using a binomial calculator or a statistical software, you can find the cumulative probability P(X ≤ 32) = 0.9955.
P(more than 32 mismatches) = 1 - P(X ≤ 32) = 1 - 0.9955 ≈ 0.0045
Therefore, the probability of more than 32 mismatches is approximately 0.0045.