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. 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.

$87\%$of the Internet sites still work within two years of publication.

$7$Internet references from scientific journal are chosen at random.

Two events are independent, if the probability of occurrence of one event does not affect the probability of occurrence of other event.

According to multiplication rule for independent events,

$P(AandB)=P(A\cap B)=P\left(A\right)\times P\left(B\right)$

Let

A: One reference still works two years later

B: $7$references still work two years later

Now,

Probability for the reference still works two years later,

$P\left(A\right)=87\%=0.87$

Since the references are selected at random, it would be more convenient to assume that references are independent of each other.

Thus,

For probability that $7$references still work two years later, apply multiplication rule for independent events:

$P\left(B\right)=P\left(A\right)\times P\left(A\right)\times ...\times P\left(A\right)=(P{\left(A\right))}^7={(0.87)}^7\approx 0.3773$

Thus,

Probability for the randomly selected all $7$references still work two years later is approx.$0.3773$.

According to complement rule,

$P\left(A^c\right)=P(notA)=1-P\left(A\right)$

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

Probability for randomly selected all $7$references still work two years later,

$P\left(B\right)\approx 0.3773$

We have of find the probability for at least $1$of the $7$references does not work two years later.

That means

None of the $7$references works two years later.

Apply the complement rule:

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

Thus,

Probability that at least $1$of the $7$references does not work two years later is$0.6227$.

Two events are independent, if the probability of occurrence of one event does not affect the probability of occurrence of other event.

According to multiplication rule for independent events,

$P(AandB)=P(A\cap B)=P\left(A\right)\times P\left(B\right)$

In Part (a),

Multiplication rule for independent events has been used.

When $7$references are chosen from one issue of same journal, we are more likely to select some references from the same website.

That means

If one of the $7$references no longer works, it is possible that other references also no longer work.

This implies

The references will be no longer independent.

Thus,

Use of the multiplication for independent events would be inappropriate.