If you meant to type: P(6 = k) = (1 – p)k–1p
The equation you've provided is not accurate. The correct expression for the probability of a binomial distribution is:
P(k) = (n choose k) * p^k * (1 - p)^(n - k)
Where:
- n is the number of trials
- k is the number of successes
- p is the probability of success in a single trial
- (n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials
BASICALLY:
- P(6 = k) refers to the probability that a specific event occurs exactly k times out of 6 trials.
- (1 - p)^(k-1) represents the probability of failure (1 - p) occurring k-1 times before the final success.
- p represents the probability of success in a single trial.
This expression is derived from the binomial distribution formula, where the number of trials (n) is fixed at 6, and we are looking at the probability of obtaining a specific number of successes (k) in those trials.
If you meant to type: P(6 - k) = (1 – p)k–1p
To simplify the expression P(6 - k) = (1 - p)^(k-1) * p, we can manipulate the exponent a little bit:
P(6 - k) = (1 - p)^(k-1) * p
Since 6 - k is k - 1 more than 6 - (k - 1), we can rewrite the expression as:
P(6 - k) = (1 - p)^(6 - (k - 1) - 1) * p
Simplifying further:
P(6 - k) = (1 - p)^(7 - k) * p
Therefore, P(6 - k) = (1 - p)^(7 - k) * p.