Question

Past experience has shown that, on the average, only 1 in 10 wells drilled hits oil. Let \(x\) be the number of drillings until the first success (oil is struck). Assume that the drillings represent independent events. a. Find \(p(1), p(2),\) and \(p(3)\). b. Give a formula for \(p(x)\). c. Graph \(p(x)\).

Short answer

Answer: The probabilities of achieving the first success in the 1st, 2nd, and 3rd drilling attempts are \(p(1)=\frac{1}{10}\), \(p(2)=\frac{9}{100}\), and \(p(3)=\frac{81}{1000}\), respectively. The general formula for the probability of achieving the first success in the x-th drilling attempt is \(p(x) = \left(\frac{9}{10}\right)^{x-1} \times \frac{1}{10}\). To graph this function, plot the function for \(x=1,2,3,\cdots\) on the x-axis representing the number of drilling attempts, and the y-axis showing the probabilities for each attempt. The graph will consist of individual points, not a continuous curve, and it will show that the probability of achieving the first success decreases as the number of attempts increases.

Step-by-step solution

Understand the concepts behind the geometric distribution

The geometric distribution is a discrete probability distribution that models the number of trials required for the first success in a sequence of Bernoulli trials with probability p. It is important to note that the trials are independent. In this case, the probability of success (hitting oil) is 1 in 10 or \(p=\frac{1}{10}\). The probability of failure (not hitting oil) is \(q=1-p=\frac{9}{10}\).

Calculate p(1), p(2), and p(3)

Using the definition of the geometric distribution, the probability of achieving the first success in the x-th trial is given by \(p(x)=q^{x-1}p\). So, we can calculate the probabilities for each case: a) \(p(1)= q^0 p= \left(\frac{9}{10}\right)^0 \times \frac{1}{10} =1 \times \frac{1}{10} = \frac{1}{10}\) b) \(p(2)= q^{2-1} p= \left(\frac{9}{10}\right)^1 \times \frac{1}{10} =\frac{9}{10} \times \frac{1}{10} = \frac{9}{100}\) c) \(p(3)= q^{3-1} p= \left(\frac{9}{10}\right)^2 \times \frac{1}{10} =\frac{81}{100} \times \frac{1}{10} = \frac{81}{1000}\)

Establish a formula for p(x)

Based on our calculations in Step 2, we can establish a general formula for \(p(x)\) following the geometric distribution. The formula is: \(p(x)=q^{x-1} p = \left(\frac{9}{10}\right)^{x-1} \times \frac{1}{10}\)

Graph p(x)

To graph the function \(p(x)\), use the formula from Step 3 as your function. Note that this is a discrete function, so its graph will consist of individual points, not a continuous curve. To create the graph, plot the function \(p(x) = \left(\frac{9}{10}\right)^{x-1} \times \frac{1}{10}\) for \(x=1,2,3,\cdots\) On the x-axis, the number of drilling attempts is represented, whereas the y-axis shows the probabilities for each attempt. Plot the points corresponding to the probabilities calculated in step 2 (\(p(1)=\frac{1}{10}, p(2)=\frac{9}{100}, p(3)=\frac{81}{1000}\)) and additional points for larger x values. The graph will show that the probability of achieving the first success decreases as the number of attempts increases.

Key concepts

Bernoulli Trials

In probability theory, a Bernoulli trial is a simple experiment with exactly two possible outcomes: typically referred to as "success" and "failure." The probability of success, denoted as \( p \), remains constant for each trial. For example, when drilling for oil, striking oil can be considered a success with a probability of \( p = \frac{1}{10} \) as given.
One essential feature of Bernoulli trials is that they are the building blocks of bigger statistical models, like the geometric distribution. This distribution applies when you wish to find the number of trials needed to get the first success. Imagine each oil drilling attempt as one Bernoulli trial; finding oil (success) has a fixed probability of \( \frac{1}{10} \).
Bernoulli trials must fulfill specific criteria:

• There are only two outcomes - success or failure.
• The probability of success does not change between trials.
• The trials are independent of each other, which we will discuss further in the 'Independent Events' section.

Probability of Success

Probability of success, denoted by \( p \), is a fundamental concept in evaluating statistical experiments. It shows how likely an event or particular outcome is. In the context of drilling for oil, the probability of striking oil with each attempt is given as \( p = \frac{1}{10} \). This means there is a 10% chance of hitting oil in any given drilling attempt.
This probability remains constant throughout all attempts. It is crucial because it helps to calculate results in distributions like the geometric distribution. In our exercise, the formula to find the probability of the first success after \( x \) trials is: \[p(x) = q^{x-1} p\] where \( q = 1 - p \) is the probability of failure.

• For \( p(1) \), or striking oil on the first attempt, the probability is \( \frac{1}{10} \).
• For \( p(2) \), or striking on the second attempt, the probability becomes \( \frac{9}{100} \).
• For \( p(3) \), or striking on the third attempt, it is \( \frac{81}{1000} \).

Looking at the formula and these numbers reveals that as attempts increase, the probability of striking oil decreases, which is a typical behavior for geometric distributions.

Independent Events

The concept of independent events is crucial when understanding Bernoulli trials and probability in general. Independent events mean that the outcome of one event does not affect the other. In our oil drilling case, this means that whether you find oil or not on one attempt does not influence the result of another attempt.
This independence is an essential aspect of modeling events with a geometric distribution. It assumes each drilling attempt is entirely separate from the others, allowing us to consistently apply the same probability of success (\( p = \frac{1}{10} \)) and failure through all trials.
If events were not independent, probabilities calculated with the geometric distribution would be inaccurate. So, it's important to remember that:

• The trials have no memory - outcomes of past trials don't affect future ones.
• The main probability values remain consistent because of this independence.
• For accurate modeling in statistics, event independence is a vital assumption.