A few statistics questions
Here are the questions:
1. The petrol stations along a road are located according to a Poisson distribution, with an average of 1 station in 10 km. Because of an oil shortage worldwide, there is a probabililty of 0.2 that a petrol station will be out of petrol.
(i) Find the probability that there is at most 1 petrol station in 15 km of the road.
(ii) Find the probability that the next 3 stations a driver encounters will be out of petrol.
A driver on this road knows that he can go another 15 km before his car runs out of petrol. Find the probability that he will be stranded on the road without petrol. Give your answer correct to 2 decimal places.
For this question, I can't solve the last part (in bold). Also, for (ii) I got the answer which is 0.008 but I don't quite know how I arrived at that so I need someone to explain that to me. Oh I can use the GC to solve this question so there's no need to go through all the formula.
2. Vehicles approaching a T-junction must either turn left or turn right. Observations by traffic engineers showed that on average, for every ten vehicles approaching the T-junction, one will turn left. It is assumed that the driver of each vehicle chooses direction independently. Out of 5 randomly chosen vehicles approaching the T-junction,
(i) find the probability that at least 3 vehicles turn right,
(ii) find the probability that exactly 4 vehicles turn right given that at least 3 vehicles turn right.
On a particular weekend, 40 randomly chosen vehicles approached the T-junction. Using a suitable approximation, find the probability that at least 38 of them turn right.
Again it's the last part I have problems with, and I can use the GC for this as well.
3. X is a binomial random variable, where the number of trials is 5 and the probability of success of each trial is p. Find the values of p if P(X=4)=0.12.
I know for question 3, it has got to do with the formula. But I still can't get the answer.
Thanks in advance if you could help me with any of these questions, I'm drowning in my homework, all 87 questions of them!