A sign on the petrol pumps of a chain of petrol stations encourages customers to have their oil checked, claiming that one out of every four cars should have its oil topped up. A sample of ten cars entering a petrol station is randomly chosen.
Let X denote the number of cars that should have their oil topped up.
a) State the probability distribution of X (including the values of all parameters)
Probability distribution: Answer
Binomial
Parameters: Answer
= Answer
and Answer
= Answer
. Enter your answer to 2 decimal places.
a) The probability distribution of X is a binomial distribution.
Parameters:
n = Number of trials = 10 (sample size)
p = Probability of success (a car needing its oil topped up) = 1/4 = 0.25 (since one out of every four cars should have its oil topped up)
Therefore, the probability distribution of X is given by:
P(X=k) = nCk * p^k * (1-p)^(n-k), where nCk represents the number of combinations of n items taken k at a time.
In this case, k can take values from 0 to 10, so the probability distribution is:
P(X=0) = 10C0 * (0.25)^0 * (0.75)^(10-0)
P(X=1) = 10C1 * (0.25)^1 * (0.75)^(10-1)
P(X=2) = 10C2 * (0.25)^2 * (0.75)^(10-2)
P(X=3) = 10C3 * (0.25)^3 * (0.75)^(10-3)
P(X=4) = 10C4 * (0.25)^4 * (0.75)^(10-4)
P(X=5) = 10C5 * (0.25)^5 * (0.75)^(10-5)
P(X=6) = 10C6 * (0.25)^6 * (0.75)^(10-6)
P(X=7) = 10C7 * (0.25)^7 * (0.75)^(10-7)
P(X=8) = 10C8 * (0.25)^8 * (0.75)^(10-8)
P(X=9) = 10C9 * (0.25)^9 * (0.75)^(10-9)
P(X=10) = 10C10 * (0.25)^10 * (0.75)^(10-10)
(Note: nCk represents "n choose k", which is the number of ways to choose k items from a set of n items, and is calculated as n! / (k! * (n-k)!))
To determine the probability distribution of X, we first need to identify the type of distribution. In this case, the situation involves randomly choosing cars and determining the number of cars that should have their oil topped up. Since each car has a fixed probability of needing its oil topped up (one out of four), we can conclude that the distribution is a binomial distribution.
A binomial distribution is characterized by two parameters: the number of trials, denoted by n, and the probability of success in each trial, denoted by p.
In this scenario, the number of trials is the number of cars we are sampling, which is ten. So, n = 10.
The probability of success, p, is given by the probability that a car should have its oil topped up, which is one out of four or 1/4. So, p = 1/4.
Therefore, the probability distribution of X is a binomial distribution with parameters n = 10 and p = 1/4.