Binomial Probability
Calculate binomial probability using P(X=k)=C(n,k)*p^k*(1-p)^(n-k): find the probability of exactly k successes in n independent trials each with success probability p.
Key Concepts
If $p$ is the probability of success and $q$ is the probability of failure in one trial of a binomial experiment, then the binomial probability of exactly $n$ successes in $m$ trials is given by ${} mC n p^n q^{m n}$.
Probability of 4 heads in 6 coin tosses: $P(\text{4 heads}) = ({} 6C 4)(\frac{1}{2})^4(\frac{1}{2})^2 = 15 \cdot \frac{1}{16} \cdot \frac{1}{4} = \frac{15}{64}$. Probability of rolling a 6 exactly twice in 5 rolls of a die: $P(\text{2 sixes}) = ({} 5C 2)(\frac{1}{6})^2(\frac{5}{6})^3 = 10 \cdot \frac{1}{36} \cdot \frac{125}{216} = \frac{1250}{7776} \approx 0.16$.
Ever wonder the odds of getting exactly 3 tails in 5 coin flips? This formula is your answer! It's designed for any experiment with exactly two outcomes, like heads or tails. It calculates the precise probability of a specific number of 'successes' happening over a set number of trials, which is super useful in games and stats.
Common Questions
What is the binomial probability formula?
The binomial probability formula is P(X=k) = C(n,k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, p is the probability of success on each trial, and C(n,k) is the binomial coefficient n!/(k!(n-k)!).
What conditions must be met to use the binomial probability model?
Four conditions: the number of trials n is fixed, each trial has exactly two outcomes (success or failure), each trial is independent of the others, and the probability of success p is constant across all trials. If any condition fails, the binomial model does not apply.
How do you calculate the probability of getting exactly 3 heads in 5 coin flips?
Use P(X=3) = C(5,3) * (0.5)^3 * (0.5)^2. C(5,3) = 10. (0.5)^3 = 0.125. (0.5)^2 = 0.25. So P(X=3) = 10 * 0.125 * 0.25 = 10 * 0.03125 = 0.3125 or 31.25%.