geometric and binomia

How is 'At least k trials' defined in Geometric probability?

Meaning: First success occurs on or after trial $k$.
Formula: $P(X \ge k) = (1 - p)^{k-1}$
Memory Aid: The exponent is $k-1$ because the first $k-1$ trials must be failures.

How is 'After k trials' defined in Geometric probability?

Meaning: First success occurs strictly after trial $k$.
Formula: P(X > k) = (1 - p)^k
Memory Aid: The exponent is $k$ because the first $k$ trials must all be failures.

How is 'By trial k' defined in Geometric probability?

Meaning: First success occurs on or before trial $k$.
Formula: $P(X \le k) = 1 - (1 - p)^k$
Memory Aid: This is the complement of 'first success after $k$.'

How is 'Within first k trials' defined in Geometric probability?

Meaning: Same as 'By trial $k$' (first success occurs on or before trial $k$).
Formula: $P(X \le k) = 1 - (1 - p)^k$
Memory Aid: 'Within' always includes trial $k$.'

How is 'At least one success in n trials' calculated?

Meaning: One or more successes occur within $n$ trials.
Formula: $1 - (1 - p)^n$
Memory Aid: Use the complement of 'zero successes' (all trials are failures).

How is 'No success in first n trials' defined?

Meaning: First success occurs after $n$ trials.
Formula: $(1 - p)^n$
Memory Aid: The exponent equals the number of trials without success.

How is 'At least k trials' defined in Geometric probability?

Meaning: First success occurs on or after trial $k$.
Formula: $P(X \ge k) = (1 - p)^{k-1}$
Memory Aid: The exponent is $k-1$ because the first $k-1$ trials must be failures.

How is 'After k trials' defined in Geometric probability?

Meaning: First success occurs strictly after trial $k$.
Formula: P(X > k) = (1 - p)^k
Memory Aid: The exponent is $k$ because the first $k$ trials must all be failures.

How is 'By trial k' defined in Geometric probability?

Meaning: First success occurs on or before trial $k$.
Formula: $P(X \le k) = 1 - (1 - p)^k$
Memory Aid: This is the complement of 'first success after $k$.'

How is 'Within first k trials' defined in Geometric probability?

Meaning: Same as 'By trial $k$' (first success occurs on or before trial $k$).
Formula: $P(X \le k) = 1 - (1 - p)^k$
Memory Aid: 'Within' always includes trial $k$.

How is 'At least one success in n trials' calculated?

Meaning: One or more successes occur within $n$ trials.
Formula: $1 - (1 - p)^n$
Memory Aid: Use the complement of 'zero successes' (all trials are failures).

How is 'No success in first n trials' defined?

Meaning: First success occurs after $n$ trials.
Formula: $(1 - p)^n$
Memory Aid: The exponent equals the number of trials without success.

What are the four conditions for a Binomial probability setup (BINS)?

1. Binary: Outcomes are classified as success or failure.
   2. Independent: Trials must be independent of one another.
   3. Number: There is a fixed number of trials ($n$).
   4. Success: The probability of success ($p$) must be the same for each trial.