The Binomial Calculator computes probabilities for binomial experiments, where each trial has exactly two outcomes (success or failure). It handles exact probabilities, cumulative probabilities, and expected value calculations.
Enter the number of trials (n), probability of success (p), and number of successes (k) to get instant results. The probability distribution bar chart shows the likelihood of each possible outcome, giving you a clear visual picture of the distribution.
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The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. It requires two parameters: n (the number of trials) and p (the probability of success on each trial).
The probability mass function is P(X = k) = C(n, k) * p^k * (1 - p)^(n - k), where C(n, k) = n! / (k!(n - k)!) is the binomial coefficient. This formula counts the number of ways to choose which k trials are successes, multiplied by the probability of that specific sequence of outcomes.
The expected value (mean) of a binomial distribution is E(X) = n * p, the variance is Var(X) = n * p * (1 - p), and the standard deviation is the square root of the variance. As n increases and p is not too close to 0 or 1, the binomial distribution approximates a normal distribution.
Aufgabe: A coin is flipped 10 times. What is the probability of getting exactly 7 heads?
Loesung: n = 10, k = 7, p = 0.5. P(X = 7) = C(10,7) * 0.5^7 * 0.5^3 = 120 * 0.0078125 * 0.125 = 0.1172.
Antwort: P(X = 7) = 0.1172 (about 11.72%)
Aufgabe: A multiple choice test has 20 questions with 4 choices each. What is the probability of getting 8 or fewer correct by guessing?
Loesung: n = 20, p = 0.25, k = 8. Sum P(X = 0) + P(X = 1) + ... + P(X = 8) using the binomial formula.
Antwort: P(X <= 8) = 0.9591 (about 95.91%)
Aufgabe: A fair die is rolled 60 times. What is the expected number of sixes?
Loesung: n = 60, p = 1/6. E(X) = 60 * (1/6) = 10. Var(X) = 60 * (1/6) * (5/6) = 8.333.
Antwort: Expected value = 10, standard deviation = 2.887
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