Overview
This lecture explains how to use the binomial distribution formula to calculate the probability that 3 out of 5 patients with a disease will survive, given a known survival rate.
Binomial Distribution Basics
- A binomial distribution applies when there are n independent trials, each with two outcomes: success or failure.
- The probability of success, denoted by p, remains constant in each trial.
- The formula for the probability of exactly r successes in n trials is:
[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} ]
Applying to the Medical Example
- 60% of patients die, so the probability of survival (success) is p = 1 - 0.6 = 0.4.
- Number of trials (patients) n = 5.
- Number of successes (survivors) r = 3.
- Substitute values into formula:
[ P(X = 3) = \binom{5}{3} (0.4)^3 (0.6)^2 ]
Calculation Steps
- Calculate combinations:
(\binom{5}{3} = \frac{5!}{3! \cdot 2!} = 10 )
- Calculate powers:
(0.4^3 = 0.064), (0.6^2 = 0.36)
- Multiply:
(10 \times 0.064 \times 0.36 = 0.2304)
- The probability three out of five survive is 0.2304 or 23.04%.
Key Terms & Definitions
- Binomial Distribution — Discrete probability distribution for n independent trials with two possible outcomes per trial.
- Success — The outcome of interest; here, a patient's survival.
- n — Number of independent trials.
- p — Probability of success in one trial.
- Combination ((\binom{n}{r})) — Number of ways to choose r successes from n trials.
Action Items / Next Steps
- Practice calculating binomial probabilities using different values of n, r, and p.