Answer: (a) Since the time to get the order follows a uniform distribution between 19 to 35 minutes, the probability of getting the order in any interval of time within this range is proportional to the length of that interval. Therefore, the probability of taking at least 23 minutes to get the order is:
P(X ≥ 23) = (35 - 23) / (35 - 19) = 0.4
Rounding to 4 decimal places, the probability is:
P(X ≥ 23) ≈ 0.4000
(b) If you arrive at Tim Horton's at 7:30am and your friends show up at 8:00am, you have 30 minutes to receive your order. Since the time to get the order follows a uniform distribution, the probability of receiving the order before your friends show up is the probability of getting the order in 30 minutes or less, which is:
P(X ≤ 30) = (30 - 19) / (35 - 19) = 0.5789
Rounding to 4 decimal places, the probability is:
P(X ≤ 30) ≈ 0.5789
(c) Let's call the time in minutes to get the order that is longer than 10% of the time as t. Since the distribution is uniform, we know that the probability of taking longer than t minutes is 0.1. Therefore, we can write:
(35 - t) / (35 - 19) = 0.1
Solving for t, we get:
t = 19 + 0.9(35 - 19) = 32.4
Rounding to 2 decimal places, we get:
t ≈ 32.40 minutes
Therefore, 10% of the time it takes longer than 32.40 minutes to get the order.
Explanation: