3. A mail order magazine service has the exclusive subscription rights to a certain magazine. Subscriptions can be reserved for one, three, four or five years. A study of the subscriber list for other similar magazines reveals that the following information regarding the probabilities of subscription length would be appropriate: there is three times as many one year subscribers as five year subscribers and the number of three year and four year subscribers are each 10 % of the total.
The mail order service company receives a fee of $3.00 for a one year subscription and another $1.00 for each additional year.
What is the probability
Chances that can happen or not
Chances
To find the probability, we need to determine the number of subscribers for each subscription length.
Let's assume that the total number of subscribers is x.
Given that there are three times as many one-year subscribers as five-year subscribers, we can say that the number of one-year subscribers is 3 times the number of five-year subscribers. So, the number of one-year subscribers would be 3y, where y is the number of five-year subscribers.
Given that the number of three-year and four-year subscribers is each 10% of the total, we can say that the number of three-year subscribers is 0.1x, and the number of four-year subscribers is also 0.1x.
Now, let's add up the number of subscribers for each subscription length:
Number of one-year subscribers = 3y
Number of three-year subscribers = 0.1x
Number of four-year subscribers = 0.1x
Number of five-year subscribers = y
Since the total number of subscribers is x, we can write the equation:
x = 3y + 0.1x + 0.1x + y
Simplifying the equation:
x = 4y + 0.2x
To solve for x, we need to eliminate the x term on the right side of the equation. We can do this by subtracting 0.2x from both sides:
x - 0.2x = 4y
0.8x = 4y
Now, let's express x in terms of y:
x = 5y
Now, we can substitute this expression for x back into our equation:
5y = 4y + 0.2(5y)
5y - 4y = y = y
So, we have found that y = 1.
Now, let's find the probabilities for each subscription length:
Probability of one-year subscription = (Number of one-year subscribers) / (Total number of subscribers)
= (3y) / (x)
= (3 * 1) / (5 * 1)
= 3/5
Probability of three-year subscription = (Number of three-year subscribers) / (Total number of subscribers)
= (0.1x) / (x)
= 0.1
Probability of four-year subscription = (Number of four-year subscribers) / (Total number of subscribers)
= (0.1x) / (x)
= 0.1
Probability of five-year subscription = (Number of five-year subscribers) / (Total number of subscribers)
= (y) / (x)
= (1) / (5)
= 1/5
Therefore, the probabilities are as follows:
Probability of one-year subscription = 3/5
Probability of three-year subscription = 0.1
Probability of four-year subscription = 0.1
Probability of five-year subscription = 1/5