Binomial Distribution

In a binomial distribution, the probability of success on each trial is denoted by p.

The probability of failure on each trial in a binomial distribution is denoted by q.

The number of trials in a binomial distribution is denoted by n.

The probability of obtaining exactly (k) successes in (n) trials in a binomial distribution is given by (P(X = k) = \binom{n}{k} p^k q^{n-k}).

The mean of a binomial distribution is given by (np).

The variance of a binomial distribution is given by (npq).

The probability of getting exactly 5 heads in 10 tosses is (P(X = 5) = \binom{10}{5} (0.5)^5 (0.5)^5 = 0.246).

The probability of getting exactly 3 sixes in 6 rolls is (P(X = 3) = \binom{6}{3} (1/6)^3 (5/6)^3 = 0.143).

The probability of getting exactly 5 correct answers in 10 questions is (P(X = 5) = \binom{10}{5} (1/4)^5 (3/4)^5 = 0.006).

The probability of receiving exactly 5 defective orders in a day is (P(X = 5) = \binom{100}{5} (0.05)^5 (0.95)^95 = 0.084).

The probability of obtaining a value between 5 and 15 is (P(5 \le X \le 15) = P(\frac{5-10}{3} \le \frac{X-10}{3} \le \frac{15-10}{3}) = P(-1.67 \le Z \le 1.67) = 0.693).

The probability of obtaining a value greater than 25 is (P(X > 25) = P(\frac{X-20}{4} > \frac{25-20}{4}) = P(Z > 1.25) = 0.023).

The probability of obtaining a value less than 30 is (P(X < 30) = P(\frac{X-50}{10} < \frac{30-50}{10}) = P(Z < -2) = 0.004).

The probability of obtaining a value between 70 and 130 is (P(70 \le X \le 130) = P(\frac{70-100}{15} \le \frac{X-100}{15} \le \frac{130-100}{15}) = P(-2 \le Z \le 2) = 0.683).

The probability of obtaining a value greater than 220 is (P(X > 220) = P(\frac{X-200}{20} > \frac{220-200}{20}) = P(Z > 1) = 0.023).