# Binomial distribution and finding the probability

I want to know how to use the binomial distribution table, given that the mean of a binomial distribution is 5 and the standard deviation is 2. What is the actual probability of 5 successes? The(mean) is found by number of trials *probability of success in any trial; while the S.D. is found by square root of (number of trials) * (probability of success) * (probability of success-1). How do i use the binomial distribution table to find the probability or even without it, but without the use of permutations and combinations.

## 2 Answers

At the first step, you must calculate the value of $p$. $$mean = \mu = np = 5 \quad(1)\\ sd = 2\Rightarrow \sqrt{np(1-p)}=2 \quad (2)$$

$$(1),(2)\Rightarrow \frac{4}{5} = 1-p\Rightarrow p = 0.2 \quad (3)$$

$$(1),(3)\Rightarrow n = 25$$ In the binomial distribution table, you must find a row with $n=25,x=5$ and column with $p=0.2$.

• You are right until the probability part. Probability of success in any trial is 0.2. But the probability of success in 5 trials is not 0.000. – Marie Anne Oct 29 '17 at 6:57
• There is no row with n=25 in the binomial table? – Marie Anne Oct 29 '17 at 7:01
• @MarieAnne You can use this site stattrek.com/online-calculator/binomial.aspx – Hasan Heydari Oct 29 '17 at 8:00
• I need to know from the table because i have an exam and we won't be able to use anything online – Marie Anne Oct 29 '17 at 8:16
• I just revisited your answer! Thank you! – Marie Anne Oct 31 '17 at 0:50

If you don´t have a table with $n=25$ and a calculator which is able to calculate binomial coefficients directly you can decompose the binomial coefficient.

In general the asked probability is $P(X=k)=\binom{n}{k} \cdot p^k\cdot (1-p)^{n-k}$

The binomial coefficient can be written as $$\binom{n}{k}=\frac{n\cdot (n-1)\cdot \ldots \cdot (n-k+1)}{1\cdot 2\cdot \ldots\cdot k}$$.

For $n=25$ and $k=5$ the probability is

$$P(X=5)=\frac{25\cdot 24\cdot 23 \cdot 22\cdot 21}{1\cdot 2\cdot 3\cdot 4\cdot 5}\cdot 0.2^5\cdot 0.8^{20}$$

This term can be calculated by the most calculators which are allowed in statistic lectures.

• @MarieAnne Nice that my answer strikes the core of your question. You´re welcome. – callculus Oct 29 '17 at 21:39