Final answer:
The question involves calculating probabilities of customer arrival times using an exponential distribution with a rate of 21 customers per hour. We apply the formula P(X < x) = 1 - e-λx to various time intervals, adjusting the times into hours for the exponential distribution's rate parameter.
Step-by-step explanation:
The student's question pertains to calculating probabilities using the exponential distribution, a common model for the time between events. Given that λ (lambda) equals 21 customers per hour, we need to calculate different probabilities for a customer's arrival time. To calculate these probabilities, we'll use the formula P(X < x) = 1 - e-λx, where X is the random variable representing time between events, x is the time period of interest, and λ is the rate parameter.
a) Probability of next customer arriving within 4 minutes
λ = 21 customers per hour so we convert 4 minutes to hours by dividing by 60: 4/60 = 1/15 hours. We get P(X < 1/15) = 1 - e-(21*1/15) = 1 - e-1.4. Calculating this, we find the probability.
b) Probability of next customer arriving within 30 seconds
We convert 30 seconds to hours: 30/3600 = 1/120 hours. Then, P(X < 1/120) = 1 - e-(21*1/120) = 1 - e-0.175 and calculate the probability.
c) Probability of next customer arriving within 7 minutes
We convert 7 minutes to hours: 7/60 hours. We calculate P(X < 7/60) = 1 - e-(21*7/60) = 1 - e-2.45 to find the probability.
d) Probability of next customer arriving within 10 minutes
Similar conversion for 10 minutes gives us 10/60 hours and P(X < 10/60) = 1 - e-(21*10/60) = 1 - e-3.5. We calculate this to find the probability.