Binomial Distribution – AP Statistics

Learning Objectives

BINS Conditions: Binary (Success/Fail), Independent trials, fixed Number of trials ($n$), constant Probability of success ($p$).

The Combinatorics $\binom{n}{k}$: Doing something exactly $k$ times can happen in many unordered permutations. We multiply success $p^k$ by failures $(1-p)^{n-k}$, scaled by the combination count.

Expected Value: The average number of successes over time is simply $E(X) = n \times p$.

Tags

BinomialDiscreteProbabilityCombinatorics

       BINOMIAL PROBABILITY MASS $P(X=k)$

       $P(k) = \binom{n}{k} p^{k} (1-p)^{n-k}$

Experiment Settings (BINS)

Number of Trials ($n$) = 10

P(Success) ($p$) = 0.50

Exact Probability Query

Target Successes $k$ = 5

EXPECTED VALUE & SPREAD

Mean $\mu = np$: 5.0

StdDev $\sigma = \sqrt{npq}$: 1.58

PROBABILITY OUTCOMES

Exact $P(X = k)$: 24.6%

Cumul. $P(X \leq k)$: 62.3%

The Normal Shift: Did you know? Because of the Central Limit Theorem, if $np \geq 10$ and $n(1-p) \geq 10$, this discrete Binomial staircase turns into the perfectly smooth Continuous Normal Distribution!