A service station owner in a country town has an average of 6 cars per hour arriving for petrol. Assuming a Poisson distribution, find the probability that 3 cars refuel in a 15 minute period.
Question
A service station owner in a country town has an average of 6 cars per hour arriving for petrol. Assuming a Poisson distribution, find the probability that 3 cars refuel in a 15 minute period.
Solution
To find the probability that 3 cars refuel in a 15-minute period, we can use the Poisson distribution formula. The formula for the Poisson distribution is:
P(x; λ) = (e^(-λ) * λ^x) / x!
Where:
- P(x; λ) is the probability of x events occurring in a given time period,
- e is the base of the natural logarithm (approximately 2.71828),
- λ is the average rate of events occurring in the given time period,
- x is the number of events we are interested in.
In this case, the average rate of cars arriving for petrol is given as 6 cars per hour. To convert this to a rate per 15 minutes, we divide by 4 (since there are 4 15-minute periods in an hour). So, the average rate of cars arriving for petrol in a 15-minute period is λ = 6/4 = 1.5.
Now, we can substitute the values into the formula:
P(3; 1.5) = (e^(-1.5) * 1.5^3) / 3!
Calculating this expression will give us the probability that exactly 3 cars refuel in a 15-minute period.
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