(a) Of the next 10 cars entering a petrol station, what is the probability that exactly three of them should have their oil topped up?
To calculate the probability of exactly three cars out of the next ten entering a petrol station needing their oil topped up, you'll need to know two key pieces of information:
1. The probability of a single car needing its oil topped up.
2. The probability of a single car not needing its oil topped up.
Let's assume that you already know the probability of a single car needing its oil topped up is p, and the probability of a single car not needing its oil topped up is q. In this case, q = 1 - p.
To find the probability of exactly three out of the next ten cars needing their oil topped up, we use the binomial probability formula:
P(X = k) = (n choose k) * p^k * q^(n - k)
where:
- P(X = k) is the probability of k cars needing their oil topped up
- n is the total number of cars (in this case, ten)
- k is the specific number of cars needing their oil topped up (in this case, three)
- (n choose k) represents the binomial coefficient, which calculates the number of possible combinations of k cars out of n total cars.
To calculate this, we substitute the appropriate values into the formula:
P(X = 3) = (10 choose 3) * p^3 * q^(10 - 3)
Now, you can substitute the value of p, the probability of a single car needing its oil topped up, to get the final answer.