Question

Wells 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 probability of finding oil on the 1st, 2nd, and 3rd drilling are as follows: - p(1) = 1/10 - p(2) = 9/100 - p(3) = 81/1000 The general formula for the probability of getting the first success on the \(x\)-th drilling in a geometric distribution is: \(p(x) = (1 - \frac{1}{10})^{x - 1} \cdot \frac{1}{10}\)

Step-by-step solution

a. Finding p(1), p(2), and p(3)

Since the probability of success (finding oil) is 1/10, and the probability of failure (not finding oil) is 9/10, we can use the geometric distribution formula to find the probabilities for the first three drillings: - p(1): Probability of finding oil on the first drilling \(p(1) = (1 - \frac{1}{10})^0 \cdot \frac{1}{10} = \frac{1}{10}\) - p(2): Probability of finding oil on the second drilling \(p(2) = (1 - \frac{1}{10})^1 \cdot \frac{1}{10} = \frac{9}{10} \cdot \frac{1}{10} = \frac{9}{100}\) - p(3): Probability of finding oil on the third drilling \(p(3) = (1 - \frac{1}{10})^2 \cdot \frac{1}{10} = \frac{81}{1000} \cdot \frac{1}{10} = \frac{81}{1000}\)

b. Formula for p(x)

The general formula for the probability of getting the first success (finding oil) on the \(x\)-th drilling in a geometric distribution is as follows: \(p(x) = (1 - p)^{x - 1} \cdot p\) In our case, the probability of success (finding oil) is \(\frac{1}{10}\), so the formula for this problem becomes: \(p(x) = (1 - \frac{1}{10})^{x - 1} \cdot \frac{1}{10}\)

c. Graph p(x)

To graph \(p(x)\), we need to define the domain of the function and plug in the values to get the respective probability values: Domain: \(x = 1, 2, 3, ...\) For each value of \(x\), plug it into the formula derived in part b, and you will have the respective probabilities: - \(p(1) = \frac{1}{10}\) - \(p(2) = \frac{9}{100}\) - \(p(3) = \frac{81}{1000}\) Continue to find more probability values as needed to build the graph. A graph of a geometric distribution decays exponentially, with higher probabilities for lower values of \(x\) and decreasing probabilities as \(x\) increases.

Key concepts

Probability Theory

Probability theory lies at the heart of statistical analysis and is pivotal for understanding various types of distributions, including the geometric distribution. It allows us to quantify the likelihood of events occurring based on assumptions or historical frequencies. To expand on the original exercise provided, probability theory involves calculating the likelihood of a particular event happening; in this case, the likelihood that drilling will strike oil. The theory assumes that each drilling is an independent trial with two possible outcomes: success (finding oil) or failure (not finding oil).

When calculating probabilities, it's important to understand the concept of randomness and how it applies to real-world scenarios. For the oil drilling example, the assumption is that each drilling has an equal probability of striking oil, which reflects the randomness of finding oil below the ground. The computation for the probabilities of the first three wells is based on this assumption of random independent trials following the laws of probability theory. Consequently, a solid grasp of probability theory facilitates a deeper comprehension of how we derive formulas for the probability of success after a certain number of trials.

Independent Events

Independent events are central to many probability distributions, including the geometric distribution depicted in this drilling exercise. Two or more events are considered independent if the occurrence of one event does not affect the probability of the other event(s) occurring. This concept is paramount in our drilling example, as it is assumed that each drilling is an independent event; the outcome of one drilling has no impact on the next.

In practice, this means that whether you find oil or not on your first drill does not influence the chances of success on your second drill or any subsequent drill. This assumption allows us to apply the geometric distribution formula in a straightforward manner. Whenever facing probability questions, it’s essential to evaluate whether events are independent or not, as this assessment will guide which probability rules to apply. The proper identification of independence in events ensures the accuracy of probability calculations and predictions.

Probability Distribution Graph

Visualizing probability distributions is essential for understanding how probabilities are spread out across different outcomes. A probability distribution graph represents the probabilities of various outcomes on a graph, which is extremely useful for illustrating concepts like the decay in probabilities in geometric distributions.

In the context of the exercise, where the domain is discrete (each drill is a separate trial), we create a graph where the x-axis represents the number of drillings, and the y-axis represents the probability of striking oil. The probabilities decrease exponentially as the number of trials increases, which is characteristic of a geometric distribution. A key feature of such a graph is its steep initial decline, reflecting that most success in a geometric distribution is likely to occur earlier in the trials.

To improve the understanding for students, a step-by-step guide on how to graph these probabilities can be instrumental. Starting with lower x-values, one would plot the corresponding probabilities determined by the geometric distribution formula. Each point on the graph corresponds to a drilling attempt and its probability of success. As students plot more points, they'll observe how the probability of striking oil decreases with each additional drilling, thus visually grasping the 'memoryless' property of the geometric distribution where past failures do not influence the likelihood of future success.