Question

Broken links Internet sites often vanish or move so that references to them can’t be followed. In fact, 87% of Internet sites referred to in major scientific journals still work within two years of publication.25 Suppose we randomly select 7 Internet references from scientific journals.

a. Find the probability that all 7 references still work two years later.

b. What’s the probability that at least 1 of them doesn’t work two years later?

c. Explain why the calculation in part (a) may not be valid if we choose 7 Internet references from one issue of the same journal.

Short answer

Part a) Probability that randomly selected all 7 references still work two years later is approx.$0.3773.$

Part b) Probability that at least 1 of the 7 references doesn't work two years later is$0.6227.$

Part c)It is not necessary that references are independent of each other.
The multiplication rule for independent events cannot be applied.

Step-by-step solution

Part a) Step 1: Given information

Within two years of publication,$87\%$ of Internet sites are still operational.

At random, $7$Internet references from scientific journals are chosen.

Part a) Step 2: Calculation

Two events are independent if the probability of one event's occurrence has no bearing on the probability of the other event's occurrence.

For independent events, the multiplication rule is as follows:

$P(\mathrm{A}\text{and}\mathrm{B})=P(\mathrm{A}\cap \mathrm{B})=P(\mathrm{A})\times P(\mathrm{B})$

Let,

A: One reference is still valid after two years.

B: Two years later, seven references are still valid.

Two years later, Probability for the reference still works.

$P(\mathrm{A})=87\%=0.87$

Because the references are chosen at random, it is more convenient to assume that they are unrelated to one another.

Thus,

Apply the multiplication rule for independent events to the probability that seven references will still work two years later:

$P(\mathrm{B})=\underset{7\text{references}}{\underbrace{P(\mathrm{A})\times P(\mathrm{A})\times \dots \times P(\mathrm{A})}}=\left(P\right(\mathrm{A}){)}^7=(0.87{)}^7\approx 0.3773$

Therefore, the Probability of the randomly selected 7 references still working two years later is approx.$0.3773.$

Part b) Step 1: Given information

Within two years of publication,$87\%$ of Internet sites are still operational.

At random, $7$Internet references from scientific journals are chosen.

Part b) Step 2: Calculation

According to the complement rule,

$P\left({\mathrm{A}}^{\mathrm{c}}\right)=P\left(\text{not}\mathrm{A}\right)=1-P(\mathrm{A})$

Let

$B$: 7 references still work two years later

$B^c$: None of the 7 references still work two years later

From Part (a),

we have,

Two years later, the probability for randomly selecting all seven references is still valid.

That means,

Two years later, none of the $7$references are still valid.

Use the complement rule to help you:

$P\left({\mathrm{B}}^{\mathrm{c}}\right)=1-P(\mathrm{B})=1-0.3773=0.6227$

Therefore, the probability that at least 1 of the 7 references does not work two years later is $0.6227$

Part c) Step 1: Given information

Within two years of publication,$87\%$of Internet sites are still operational.

At random, $7$Internet references from scientific journals are chosen.

Part c) Step 2: Calculation

Two events are independent if the probability of one event's occurrence has no bearing on the probability of the other event's occurrence.

For independent events, the multiplication rule is as follows:

$P(\mathrm{A}\cap \mathrm{B})=P(\mathrm{A}\text{and}\mathrm{B})=P(\mathrm{A})\times P(\mathrm{B})$

In part (a)

For independent events, the multiplication rule was used.

We are more likely to choose some references from the same website when 7 references are chosen from one issue of the same journal.

That means,

If one of the $7$references stops working, it's possible that other references will stop working as well.

This implies

The references will no longer be self-contained.

Therefore, use of the multiplication for independent events would be inappropriate.