# 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.

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 donĀ“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