Upload Assignment: Assignment 3B.4
INSTRUCTIONS
1. The United Package Service has a policy that states if a package is delivered late they will reimburse their shipping fees. They deliver packages late only $5 \%$ of the time. If a delivery person has 7 deliveries one day, what is the probability that at most 1 package is arriving late? Round your answer to the nearest ten-thousandth.
2. A coin is being flipped 400 times. What is the probability of it landing on heads at most 170 times, rounded to the nearest tenth of a percent?
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Step 1 ：Use the binomial probability formula to find the probability of at most 1 package arriving late out of 7 deliveries: \(P(X \leq 1) = P(X=0) + P(X=1)\)

Step 2 ：Calculate \(P(X=0)\) and \(P(X=1)\) using the formula: \(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\)

Step 3 ：Plug in the values: \(n1 = 7\), \(p1 = 0.05\)

Step 4 ：Calculate the probabilities: \(P(X=0) = \binom{7}{0} (0.05)^0 (1-0.05)^{7-0} = 0.69834\) and \(P(X=1) = \binom{7}{1} (0.05)^1 (1-0.05)^{7-1} = 0.25728\)

Step 5 ：Add the probabilities: \(P(X \leq 1) = 0.69834 + 0.25728 = 0.95562\)

Step 6 ：\(\boxed{\text{The probability that at most 1 package is arriving late is 0.95562}}\)

Step 7 ：Use the binomial probability formula to find the probability of getting at most 170 heads out of 400 coin flips: \(P(Y \leq 170) = \sum_{k=0}^{170} P(Y=k)\)

Step 8 ：Calculate \(P(Y=k)\) using the formula: \(P(Y=k) = \binom{n}{k} p^k (1-p)^{n-k}\)

Step 9 ：Plug in the values: \(n2 = 400\), \(p2 = 0.5\)

Step 10 ：Calculate the probability: \(P(Y \leq 170) = 0.001564508063407257\)

Step 11 ：Round the probability to the nearest tenth of a percent: \(0.001564508063407257 \approx 0.2\%\)

Step 12 ：\(\boxed{\text{The probability of the coin landing on heads at most 170 times is 0.2\%}}\)