Solved: Approximately 30% of obese patients develop diabetes. If a p…

Approximately 30% of obese patients develop diabetes. If a physician sees 10 patients who are obese, what is the probability that half of them will develop diabetes?

 

solution

**Introduction**

The probability that half of 10 obese patients will develop diabetes can be calculated using the binomial distribution. The binomial distribution is used to model the probability of a certain number of successes in a fixed number of independent Bernoulli trials. In this case, the trials are whether each patient develops diabetes or not, and the probability of success (developing diabetes) is 30%.

**Calculation**

To calculate the probability, we need to use the formula for the binomial distribution:

P(X=k) = C(n, k) * p^k * (1-p)^(n-k)

Where:
P(X=k) is the probability of having exactly k successes,
n is the number of trials,
k is the number of successes,
p is the probability of success in each trial,
C(n, k) is the number of combinations of n items taken k at a time.

For this problem, n = 10 (number of patients), k = 5 (half of the patients), and p = 0.30 (probability of a patient developing diabetes).

Using this formula, we can now calculate the probability.

**Calculation Steps**

1. Calculate the number of combinations:
C(n, k) = C(10, 5) = 10! / (5! * (10-5)!)

2. Calculate the probability:
P(X=5) = C(10, 5) * 0.30^5 * (1-0.30)^(10-5)

**Solution**

Utilizing the above formula and calculations, the probability that half of the 10 obese patients will develop diabetes is:

P(X=5) = C(10, 5) * 0.30^5 * (1-0.30)^(10-5)

P(X=5) = 252 * 0.30^5 * 0.70^5

P(X=5) = 252 * 0.00243 * 0.16807

P(X=5) ≈ 0.102518

Therefore, the probability that exactly 5 out of the 10 obese patients will develop diabetes is approximately 0.102518, or 10.25%.

**Conclusion and Implications**

In conclusion, using the binomial distribution, we have determined that the probability of exactly half of the 10 obese patients developing diabetes is approximately 0.102518, or 10.25%. This calculated probability is based on the assumption that the probability of developing diabetes remains constant for each patient and that the patients are independent of each other.

Understanding these probabilities can help physicians, researchers, and policymakers in assessing the likelihood of diabetes development in obese patients. It can also aid in making informed decisions about the allocation of resources for prevention and treatment interventions.

It is essential to note that individual cases may not always align with the calculated probability due to inherent variations. Nonetheless, the use of probability and statistical modeling provides a valuable framework for decision-making and understanding the likelihood of specific outcomes.

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