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