Concept: The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials.
The formula for probability is:
\( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \)
- n = number of trials
- p = probability of success
Use the sliders below to explore how n and p affect the distribution shape.
Practice Question: Applying the Binomial Formula
Based on the scenario from the explanation, try to solve this problem using the formula.
Question: A new drug has a 60% cure rate. If 20 patients take the drug, what is the probability that exactly 15 of them will be cured?
- 0.015
- 0.075
- 0.124
- 0.250
Answer and Explanation
Correct Answer: B) 0.075
Explanation & Calculation:
To solve this, we use the binomial probability formula:
\(P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}\)
Here are our known values from the problem:
- n (number of trials) = 20 patients
- k (number of successes) = 15 patients
- p (probability of success) = 0.60 (60% cure rate)
- (1-p) (probability of failure) = 1 - 0.60 = 0.40
Now, we substitute these values into the formula:
\(P(X=15) = \binom{20}{15} (0.60)^{15} (0.40)^{20-15}\)
\(P(X=15) = \binom{20}{15} (0.60)^{15} (0.40)^5\)
- Calculate the binomial coefficient:
\(\binom{20}{15} = \frac{20!}{15!5!} = 15,504\)
- Calculate the probability terms:
\((0.60)^{15} \approx 0.0004701\)
\((0.40)^5 = 0.01024\)
- Multiply the results:
\(P(X=15) = 15,504 \times 0.0004701 \times 0.01024 \approx 0.07464\)
Rounding to three decimal places, the probability is **0.075**.