Bernoulli Trials

Overview

A Bernoulli trial is a random experiment with exactly two possible outcomes: "success" or "failure." The probability of success is denoted by $p$, and the probability of failure is $q = 1 - p$.

Bernoulli trials form the basis of the Binomial distribution, which models the number of successes in nn independent Bernoulli trials.

Characteristics of Bernoulli Trials

1. Two Outcomes: Each trial results in either a success or a failure.
2. Fixed Probability: The probability of success ($p$) remains the same for all trials.
3. Independence: The outcome of one trial does not affect another.

Key Formulas

Probability of Exactly $k$ Successes in $n$ Trials:

$$P(X = k) = \binom{n}{k} p^k q^{n-k}$$

Where:

• $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient.
• $k$: Number of successes.
• $n$: Total number of trials.
• $p$: Probability of success.
• $q$: Probability of failure $(q = 1-p)$

Probability that the First Success Occurs on the $n$-th Trial:

$$P(X = n) = q^{n-1} \times p$$

Probability that the kkth Success Occurs on the $n$-th Trial:

$$P(\text{kth success at } n) = \binom{n-1}{k-1} p^k q^{n-k}$$

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Worked Examples

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Summary

• Bernoulli Trial: A random experiment with two outcomes: success ($p$) and failure ($q = 1-p$).
• Key Formulas:
  • $P(X = k) = \binom{n}{k} p^k q^{n-k}$
  • $P(X = n) = q^{n-1} \times p$
  • $P(\text{kth success at } n) = \binom{n-1}{k-1} p^k q^{n-k}$
• Applications: Tossing coins, sports statistics, quality control in factories, etc.