One-fourth of all the cattle in a very large herd are infected with hoof-and-mouth disease. A government inspector will test up to 5 cattle for this disease and stops and quarantines the herd if any of the 5 selected test positive for hoof-and-mouth.

Assume the following conditions:
- The presence of the disease in any cow is independent of its presence in any other cow.
- The cattle are randomly mixed together.
- The inspector makes her selection randomly.
- The herd is sufficiently large so that the random variable \( x \), counting the number of trials necessary to find a cow with hoof-and-mouth, is geometric.

What is the probability the inspector does not find a single cow with the disease?

To solve this problem, we need to consider that the probability of picking a cow that is infected (success) is 1/4, which is also the probability (p) of having a success on each individual trial.

The converse, the probability of picking a cow that is not infected (failure), is 1-p = 3/4.

Given that the question is asking for the probability that the inspector does not find a single cow with the disease in up to 5 trials, we would like to know the probability of 5 consecutive failures.

Since the inspector’s selection are independent, the probability of several independent events happening together is the product of their individual probabilities:

P(5 failures) = (3/4)*(3/4)*(3/4)*(3/4)*(3/4) = (3/4)5 = 0.237

So, the probability that the inspector does not find a single cow with the disease in 5 trials is 0.237 or 23.7%.

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