How do you use the binomial distribution table?

Fix n and p; the row is the number of successes k. Each cell gives P(X = k) or the cumulative P(X ≤ k), depending on the tab selected.

What is the difference between P(X = k) and P(X ≤ k)?

P(X = k) is the probability of exactly k successes; P(X ≤ k) is the cumulative one, the sum of the probabilities from 0 up to k.

When should you approximate the binomial table by the normal or the Poisson?

With large n and moderate p, the binomial is approximated by the normal table; with large n and small p (λ = n·p moderate), by the Poisson table.

Binomial Distribution Table (Cumulative)

How to read the table: each cell shows P(X = k) = C(n,k)·p^k·(1−p)^(n−k). The parameter n is the number of trials, and you can choose it from the dropdown. Rows are the values of k (0, 1, …, n); columns are the values of p. Use the buttons to change the range of p.

n (trials):

Click a cell to read the value.

P(X = k) — p from 0.05 to 0.50

k	p = 0.05	p = 0.10	p = 0.15	p = 0.20	p = 0.25	p = 0.30	p = 0.35	p = 0.40	p = 0.45	p = 0.50
0	0.5987	0.3487	0.1969	0.1074	0.0563	0.0282	0.0135	0.0060	0.0025	0.0010
1	0.3151	0.3874	0.3474	0.2684	0.1877	0.1211	0.0725	0.0403	0.0207	0.0098
2	0.0746	0.1937	0.2759	0.3020	0.2816	0.2335	0.1757	0.1209	0.0763	0.0439
3	0.0105	0.0574	0.1298	0.2013	0.2503	0.2668	0.2522	0.2150	0.1665	0.1172
4	0.0010	0.0112	0.0401	0.0881	0.1460	0.2001	0.2377	0.2508	0.2384	0.2051
5	0.0001	0.0015	0.0085	0.0264	0.0584	0.1029	0.1536	0.2007	0.2340	0.2461
6	0.0000	0.0001	0.0012	0.0055	0.0162	0.0368	0.0689	0.1115	0.1596	0.2051
7	0.0000	0.0000	0.0001	0.0008	0.0031	0.0090	0.0212	0.0425	0.0746	0.1172
8	0.0000	0.0000	0.0000	0.0001	0.0004	0.0014	0.0043	0.0106	0.0229	0.0439
9	0.0000	0.0000	0.0000	0.0000	0.0000	0.0001	0.0005	0.0016	0.0042	0.0098
10	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0001	0.0003	0.0010

k	p = 0.55	p = 0.60	p = 0.65	p = 0.70	p = 0.75	p = 0.80	p = 0.85	p = 0.90	p = 0.95
0	0.0003	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
1	0.0042	0.0016	0.0005	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
2	0.0229	0.0106	0.0043	0.0014	0.0004	0.0001	0.0000	0.0000	0.0000
3	0.0746	0.0425	0.0212	0.0090	0.0031	0.0008	0.0001	0.0000	0.0000
4	0.1596	0.1115	0.0689	0.0368	0.0162	0.0055	0.0012	0.0001	0.0000
5	0.2340	0.2007	0.1536	0.1029	0.0584	0.0264	0.0085	0.0015	0.0001
6	0.2384	0.2508	0.2377	0.2001	0.1460	0.0881	0.0401	0.0112	0.0010
7	0.1665	0.2150	0.2522	0.2668	0.2503	0.2013	0.1298	0.0574	0.0105
8	0.0763	0.1209	0.1757	0.2335	0.2816	0.3020	0.2759	0.1937	0.0746
9	0.0207	0.0403	0.0725	0.1211	0.1877	0.2684	0.3474	0.3874	0.3151
10	0.0025	0.0060	0.0135	0.0282	0.0563	0.1074	0.1969	0.3487	0.5987

How to read the table: each cell shows P(X ≤ k) = Σᵢ₌₀ᵏ C(n,i)·p^i·(1−p)^(n−i). The parameter n is the number of trials, and you can choose it from the dropdown; rows indicate k and columns indicate p. To calculate P(X > k) use the complement: 1 − P(X ≤ k).

n (trials):

Click a cell to read the value.

P(X ≤ k) — p from 0.05 to 0.50

k	p = 0.05	p = 0.10	p = 0.15	p = 0.20	p = 0.25	p = 0.30	p = 0.35	p = 0.40	p = 0.45	p = 0.50
0	0.5987	0.3487	0.1969	0.1074	0.0563	0.0282	0.0135	0.0060	0.0025	0.0010
1	0.9139	0.7361	0.5443	0.3758	0.2440	0.1493	0.0860	0.0464	0.0233	0.0107
2	0.9885	0.9298	0.8202	0.6778	0.5256	0.3828	0.2616	0.1673	0.0996	0.0547
3	0.9990	0.9872	0.9500	0.8791	0.7759	0.6496	0.5138	0.3823	0.2660	0.1719
4	0.9999	0.9984	0.9901	0.9672	0.9219	0.8497	0.7515	0.6331	0.5044	0.3770
5	1.0000	0.9999	0.9986	0.9936	0.9803	0.9527	0.9051	0.8338	0.7384	0.6230
6	1.0000	1.0000	0.9999	0.9991	0.9965	0.9894	0.9740	0.9452	0.8980	0.8281
7	1.0000	1.0000	1.0000	0.9999	0.9996	0.9984	0.9952	0.9877	0.9726	0.9453
8	1.0000	1.0000	1.0000	1.0000	1.0000	0.9999	0.9995	0.9983	0.9955	0.9893
9	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	0.9999	0.9997	0.9990
10	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

k	p = 0.55	p = 0.60	p = 0.65	p = 0.70	p = 0.75	p = 0.80	p = 0.85	p = 0.90	p = 0.95
0	0.0003	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
1	0.0045	0.0017	0.0005	0.0001	0.0000	0.0000	0.0000	0.0000	0.0000
2	0.0274	0.0123	0.0048	0.0016	0.0004	0.0001	0.0000	0.0000	0.0000
3	0.1020	0.0548	0.0260	0.0106	0.0035	0.0009	0.0001	0.0000	0.0000
4	0.2616	0.1662	0.0949	0.0473	0.0197	0.0064	0.0014	0.0001	0.0000
5	0.4956	0.3669	0.2485	0.1503	0.0781	0.0328	0.0099	0.0016	0.0001
6	0.7340	0.6177	0.4862	0.3504	0.2241	0.1209	0.0500	0.0128	0.0010
7	0.9004	0.8327	0.7384	0.6172	0.4744	0.3222	0.1798	0.0702	0.0115
8	0.9767	0.9536	0.9140	0.8507	0.7560	0.6242	0.4557	0.2639	0.0861
9	0.9975	0.9940	0.9865	0.9718	0.9437	0.8926	0.8031	0.6513	0.4013
10	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000	1.0000

How to use these tables

Before looking up a value, identify the three pieces of information in the problem: the number of trials n, the probability of success p on each trial, and the number of successes k.

Choose n from the “n (trials)” dropdown. The table will update and display rows from k = 0 through k = n.
Select the appropriate tab: use P(X = k) for the probability of exactly k successes or P(X ≤ k) for the probability of at most k successes.
Choose the p range with the p ≤ 0.50 or p > 0.50 buttons, then find the column for the probability of success.
Find row k. Its intersection with column p is the requested probability; you can also click the cell to see the value identified.
Combine values when needed: for P(X > k), subtract the cumulative probability from 1; for an interval P(a ≤ X ≤ b), subtract P(X ≤ a − 1) from P(X ≤ b).

Example: for 10 trials with p = 0.30, to find the probability of exactly 4 successes, choose n = 10, open P(X = k), and use the cell where row k = 4 meets column p = 0.30.

Binomial Distribution

A binomial experiment consists of n independent trials, each with probability p of success. The number of successes X follows a B(n, p) distribution with probability mass function:

\( P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \ldots, n \)

The mean is μ = np and the variance σ² = np(1 − p).

Useful properties

Symmetry: P(X = k; n, p) = P(X = n − k; n, 1 − p). Use this for p > 0.5 by looking at the equivalent column with p' = 1 − p.
Complement: P(X > k) = 1 − P(X ≤ k).
Interval: P(a ≤ X ≤ b) = P(X ≤ b) − P(X ≤ a − 1).
For large n and small p, the binomial is well approximated by a Poisson distribution with λ = np.
For large n, the binomial is well approximated by a Normal distribution N(np, np(1−p)).

Related tables and calculators

Frequently asked questions

How do you use the binomial distribution table? Fix n and p; the row is the number of successes k. Each cell gives P(X = k) or the cumulative P(X ≤ k), depending on the tab selected.
What is the difference between P(X = k) and P(X ≤ k)? P(X = k) is the probability of exactly k successes; P(X ≤ k) is the cumulative one, the sum of the probabilities from 0 up to k.
When should you approximate the binomial table by the normal or the Poisson? With large n and moderate p, the binomial is approximated by the normal table; with large n and small p (λ = n·p moderate), by the Poisson table.