Binomial Distribution 7Bb584

1. **Problem Statement:** We want to understand the binomial distribution, which models the number of successes in a fixed number of independent trials, each with the same probability of success. 2. **Formula:** The probability of exactly $k$ successes in $n$ trials is given by the binomial probability formula: $$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where: - $n$ is the number of trials, - $k$ is the number of successes, - $p$ is the probability of success on a single trial, - $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient, representing the number of ways to choose $k$ successes from $n$ trials. 3. **Important Rules:** - Trials are independent. - Each trial has only two outcomes: success or failure. - Probability of success $p$ is constant for each trial. 4. **Step-by-step example:** Suppose we toss a fair coin $5$ times ($n=5$), and want the probability of getting exactly $3$ heads ($k=3$). The probability of success (head) is $p=0.5$. 5. Calculate the binomial coefficient: $$\binom{5}{3} = \frac{5!}{3!2!} = \frac{120}{6 \times 2} = \frac{120}{12} = 10$$ 6. Calculate the probability: $$P(X=3) = 10 \times (0.5)^3 \times (1-0.5)^{5-3} = 10 \times 0.125 \times 0.25$$ 7. Simplify: $$P(X=3) = 10 \times 0.125 \times 0.25 = 10 \times 0.03125 = 0.3125$$ 8. **Interpretation:** There is a $31.25\%$ chance of getting exactly $3$ heads in $5$ tosses. 9. **Memorization tips:** - Remember the formula structure: combinations times success probability to the power of successes times failure probability to the power of failures. - Understand the meaning of each term. - Practice with different values of $n$, $k$, and $p$. This completes the explanation and example of the binomial distribution.