Repeated Trials and Binomial Distribution in Probability

Learn about repeated trials and the binomial distribution in probability theory. This guide explains the characteristics of repeated trial experiments, including independent trials and constant probabilities, and shows how to calculate probabilities using the binomial probability formula.

Repeated Trials and Binomial Distribution in Probability

Repeated Trials

In probability, a repeated trial experiment involves repeating the same experiment multiple times. Each trial is independent, meaning the outcome of one trial doesn't influence the outcome of another. The probability of success (p) and failure (q = 1 - p) remain constant for each trial.

Example: Coin Tosses

Tossing a fair coin three times is a repeated trial experiment. The probability of getting a head (H) is 1/2, and the probability of getting a tail (T) is also 1/2.

(The possible outcomes of three coin tosses and the calculation of the probability of getting exactly two heads and one tail are given in the original text and should be included here.)

Probability of r Successes in n Trials

If a trial is repeated n times, and the probability of success is p, the probability of getting exactly r successes is given by:

P(X = r) = C(n, r) * pr * qn-r

Where:

  • C(n, r) is the number of combinations of choosing r items from a set of n items (n! / (r! * (n-r)!)).
  • p is the probability of success.
  • q is the probability of failure (q = 1 - p).

Binomial Distribution

The binomial distribution is a probability distribution that describes the probability of getting a certain number of successes in a fixed number of independent trials. The formula is:

P(X = r) = C(n, r)prqn-r

Where n is the number of trials, and r is the number of successes.

Calculating Probabilities Using the Binomial Distribution

(The calculations for probabilities of 0, 1, 2, and n successes are shown in the original text and should be included here. The proof that the sum of probabilities is always 1 should also be included.)

Examples: Applying the Binomial Distribution

Example 1: Defective Pencils

(This example, calculating probabilities related to defective pencils, is provided in the original text and should be included here. The calculations for the probabilities of exactly three defective pencils, at least two defective pencils, and no defective pencils should be clearly shown.)

Example 2: Coin Tosses

(This example, calculating the probability of getting exactly 5 heads in 15 coin tosses, is provided in the original text and should be included here. The solution should be clearly shown.)

Example 3: Expected Number of Cases

(This example, calculating the expected number of cases with 10 heads and 6 tails in 256 sets of 16 coin tosses, is provided in the original text and should be included here. The solution should be clearly shown.)

Example 4: Expected Number of Cases

(This example, calculating the expected number of cases with 4 heads and 2 tails in 16 sets of 6 coin tosses, is provided in the original text and should be included here. The solution should be clearly shown.)

Example 5: Defective Parts in Sampling

(This example, calculating the expected number of samples with at least two defective parts, is provided in the original text and should be included here. The solution should be clearly shown.)

Conclusion

Repeated trials and the binomial distribution are fundamental concepts in probability theory, providing a framework for analyzing experiments with multiple independent trials and a constant probability of success.