Questions: Use the binomial distribution below to answer the following question(s). A doctor knows from experience that 10% of the patients to whom she gives a certain medication will experience undesirable side effects. Assume the doctor gives medication to the next 12 patients. 1) Referring to the binomial distribution for the medication, find the probability that exactly three of these patients will experience undesirable side effects. P(x) = n C x * p^x * q^(n-x)

Use the binomial distribution below to answer the following question(s). A doctor knows from experience that 10% of the patients to whom she gives a certain medication will experience undesirable side effects. Assume the doctor gives medication to the next 12 patients.
1) Referring to the binomial distribution for the medication, find the probability that exactly three of these patients will experience undesirable side effects.
P(x) = n C x * p^x * q^(n-x)

Transcript text: Use the binomial distribution below to answer the following question(s). A doctor knows from experience that $10 \%$ of the patients to whom she gives a certain medication will experience undesirable side effects. Assume the doctor gives medication to the next 12 patients. 1) Referring to the binomial distribution for the medication, find the probability that exactly three of these patients will experience undesirable side effects. \[ P(x)={ }_{n} C_{x} \cdot p^{x} \cdot q^{n-x} \]

Solution

Solution Steps

Step 1: Define the Problem

We are tasked with finding the probability that exactly $ k = 3 $ out of $ n = 12 $ patients will experience undesirable side effects from a medication, where the probability of experiencing side effects for each patient is $ p = 0.10 $.

Step 2: Use the Binomial Probability Mass Function

The probability of exactly $ k $ successes in $ n $ trials for a binomial distribution can be calculated using the formula:

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

In our case, we need to compute:

\[ P(X = 3) = {12 \choose 3} \cdot (0.10)^3 \cdot (0.90)^{12 - 3} \]

Step 3: Calculate the Probability

After performing the calculations, we find that the probability $ P(X = 3) $ is approximately $ 0.0852 $.