Binomial Probability Distributions (6.2)

Only two outcomes per trial (success or failure).
Trials are independent (no dependencies across trials).
Fixed number of trials n (the total number of experiments).
Constant probability of success p on each trial; probability of failure is q = 1 − p.
Random variable X = number of successes in n trials.
Probability of exactly x successes is given by the binomial formula: P(X=x)= \binom{n}{x} p^{x} q^{\,n-x} with q=1-p (or P(X=x)= \binom{n}{x} p^{x} (1-p)^{n-x}).

A percent (or probability) is given for a single trial, often tied to success.
There are two outcomes (success vs. not success).
The number of trials is fixed and does not depend on results (independence).
If you see words like “exactly x,” “not more than,” or “at least,” you’ll usually form binomial calculations.

If the problem uses a variable total number of trials (e.g., "until" a certain event occurs with no fixed end), it may not be binomial.
Example note: rolling a die until a 5 appears exactly 11 times is not binomial if the total trials aren’t fixed in advance.

For exact x successes, use the Binomial PDF program: Binomial PDF(n, p, x).
Steps (typical calculator workflow):
- Access distribution functions: 2nd VARS (distribution) → Binomial PDF.
- Enter in order: n, p, x (careful with placement: n first, then p, then x).
- Run and record the probability, usually to four decimal places.
In many problems you will also need the formula form: P(X=x)=\binom{n}{x} p^{x} (1-p)^{n-x}, but the program handles the calculation.
The program is for exactly x successes. For “at least” or “at most,” compute multiple P(X=x) values and sum them (or compute separately and add).

If a problem provides a percent for a single-trial success, you’ll almost always be in a binomial setup.
Always identify these four pieces before calculator work: A) what is a success, B) n (total trials), C) p (probability of success per trial), D) x (number of successes).
Complement rule: q = 1 − p; you may need q for intermediate steps or other parts of a problem.
Unusual (rare) event cutoff: a probability < 0.05 is considered unusual.
When rounding, follow the problem’s instruction (e.g., four decimal places) and be consistent when combining results.

Example 1: Archer hits bull’s-eye with probability p = 0.78; n = 18 trials; find P(X=13).
- X = number of bull’s-eyes; X ~ Binomial(n=18, p=0.78).
- P(X=13) = \binom{18}{13} (0.78)^{13} (0.22)^{5} ≈ 0.0175 (four decimals).
Example 2: Seed germination problem—n = 120, p = 0.82, find P(X=93).
- X ~ Binomial(n=120, p=0.82).
- p is given, derive q = 1 − p = 0.18; use Binomial PDF with n, p, x.
Example 3 (overbooking): 20 booked, 19 seats, p = 0.85 (probability a booked passenger arrives).
- Not enough seats means X = 20 arrivals out of n = 20 trials.
- P(X=20) = \binom{20}{20} (0.85)^{20} (0.15)^{0} = (0.85)^{20} \approx 0.0388
- Interpretation: probability flight is full (unusual if < 0.05).

Identify binomial structure: two outcomes, independence, fixed n, constant p.
Determine n, p, x, q = 1 − p.
Use Binomial PDF for P(X=x): P(X=x)=\binom{n}{x} p^{x} q^{n-x}.
If the problem asks for at least/at most, sum the relevant P(X=x) values.
Use the calculator to avoid manual calculation errors; show setup: program name and the inputs (n, p, x).
Compare results to 0.05 to define unusual events.