A sign on the gas pumps of a chain of gasoline stations encourages customers to have their oil checked, claiming that one out of four cards need to have oil added. If this is true, what is the probability of the following events?
a. One out of the next four cars needs oil.
b. Two out of the next eight cars need oil.
c. Three out of the next 12 cars need oil.
I can not figure out how to set this up mathematically.
To calculate the probabilities of these events, we need to use the concept of probability and the information provided. Let's walk through the process step by step.
Step 1: Understand the problem
The sign claims that one out of four cars needs oil. This tells us that the probability of a car needing oil is 1/4.
Step 2: Define the events
a. Event A: One out of the next four cars needs oil.
b. Event B: Two out of the next eight cars need oil.
c. Event C: Three out of the next 12 cars need oil.
Step 3: Calculate the probabilities
a. Event A: One out of the next four cars needs oil.
To calculate the probability, we need to use the concept of "and" probabilities. In this case, we want one car to need oil and the other three cars not to need oil. Since each car is independent of each other, we can multiply the individual probabilities.
Probability of one car needing oil = 1/4
Probability of three cars not needing oil = (3/4) * (3/4) * (3/4) = 27/64
Probability of Event A = Probability of one car needing oil * Probability of three cars not needing oil
= (1/4) * (27/64)
= 27/256
b. Event B: Two out of the next eight cars need oil.
Similar to Event A, now we want two cars to need oil and the other six cars not to need oil. Again, since each car is independent, we can multiply the individual probabilities.
Probability of two cars needing oil = (1/4) * (1/4)
Probability of six cars not needing oil = (3/4) * (3/4) * (3/4) * (3/4) * (3/4) * (3/4) = 729/4096
Probability of Event B = Probability of two cars needing oil * Probability of six cars not needing oil
= (1/4) * (1/4) * (729/4096)
= 729/65536
c. Event C: Three out of the next twelve cars need oil.
Again, using the same approach, we want three cars to need oil and the other nine cars not to need oil. We can calculate the probabilities as follows:
Probability of three cars needing oil = (1/4) * (1/4) * (1/4)
Probability of nine cars not needing oil = (3/4)^9 = 19683/262144
Probability of Event C = Probability of three cars needing oil * Probability of nine cars not needing oil
= (1/4) * (1/4) * (1/4) * (19683/262144)
= 19683/4194304
So, the probabilities of the events are:
a. Probability of Event A = 27/256
b. Probability of Event B = 729/65536
c. Probability of Event C = 19683/4194304