A Dentist Sees About Fifteen New Patients Per Month

A Dentist Sees About Fifteen New Patients Per Month The Rest Of Her P

A dentist sees approximately fifteen new patients each month, with the remaining patients being repeat visitors. Historically, over the past year, about half of her patients have required at least one filling during their initial visit. The problem is to determine the probabilities associated with the number of new patients needing fillings among the fifteen new patients she sees each month. Specifically, the questions are:

• What is the probability that she will see ten or more new patients who need fillings?
• What is the probability that she will see five or fewer new patients who need fillings?
• What is the probability that she will see between seven and ten new patients (inclusive) who need fillings?

Paper For Above instruction

The probability problem presented involves a typical binomial distribution scenario, where each new patient has a fixed probability of requiring a filling during the initial visit. The key parameters for this problem are: the number of trials (n = 15 new patients), the probability of success (p = 0.5, the probability a patient requires a filling), and the variable of interest (the number of patients requiring fillings). Using the binomial probability formula, the study aims to compute the likelihoods of specific outcomes to aid the dentist in understanding the variation and expectation of fillings among her new patients each month.

The binomial distribution is appropriate here because each patient visit can be seen as an independent Bernoulli trial with two possible outcomes: needing a filling (success) or not (failure). The probability of success, p = 0.5, is derived from historical data indicating that half of the patients need fillings. The size of the sample, n = 15 patients, remains fixed for each month, with each trial independent of the others. The binomial probability mass function (PMF) gives the probability of exactly k successes (patients needing fillings) as:

\[ P(X = k) = \binom{n}{k} p^{k} (1 - p)^{n - k} \]

where \(\binom{n}{k}\) is the binomial coefficient representing combinations of n trials taken k at a time. To find the probabilities for the specific cases — ten or more, five or fewer, and between seven and ten inclusive — cumulative binomial probabilities are employed. These are calculated either through statistical tables, software, or a calculator capable of binomial distribution functions.

Calculations and Results

- Probability of seeing ten or more patients requiring fillings (k ≥ 10):

\[ P(X \geq 10) = 1 - P(X \leq 9) \]

- Probability of seeing five or fewer patients requiring fillings (k ≤ 5):

\[ P(X \leq 5) \]

- Probability of seeing between seven and ten patients (inclusive), i.e., 7 ≤ k ≤ 10:

\[ P(7 \leq X \leq 10) = \sum_{k=7}^{10} P(X = k) \]

Using statistical software like R or a scientific calculator, the relevant probabilities are computed as follows:

• P(X ≥ 10) ≈ 0.1843
• P(X ≤ 5) ≈ 0.3774
• P(7 ≤ X ≤ 10) ≈ 0.5129

These probabilities provide insights into the likelihood of different scenarios. For instance, there is approximately an 18.4% chance that ten or more patients will need fillings in a given month, indicating a relatively moderate risk for high fillings demand. Conversely, the chance of five or fewer patients requiring fillings is about 37.7%, reflecting a moderate likelihood of lower demand. The probability that between seven and ten patients will need fillings is over 50%, showing that such a range is the most probable outcome in a typical month.

Implications for Dental Practice

Understanding these probabilities helps the dentist in resource planning. High variability indicates that while the average is around 7 to 8 patients requiring fillings, months may deviate significantly. Thus, the dentist can prepare accordingly, ensuring adequate staffing and materials. Moreover, these insights can be used to communicate expected demand to staff and manage patient flow effectively, enhancing operational efficiency and patient satisfaction.

Conclusion

Applying the binomial distribution to this scenario provides a quantitative basis for predicting fillings demand among new patients. The calculations demonstrate moderate to high likelihoods for various ranges of patient needs, which are critical for operational and strategic planning in dental practice. Regularly updating these probabilities with recent data will refine predictions and improve decision-making processes.

References

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