Binomial distribution and Number of Successes

Question

I was wondering if there is a formula relating the number of successes and the sample size in a binomial distribution. Is there such a formula?

Answer 1

I'm not quite sure at what level you are asking the question, but at it's most simple:

The most likely number of successes, $s$ depends upon the probability of success, $p$ as well as the sample size, $n$. This crucial dependency is missing altogether from the question as asked.

If I toss a biased coin 200 times with a probability of landing Heads (success) of 0.3 then the most likely number of successes is 60.

So, the sample size was 200, the probability of success was 0.3, and the most likely number of successes 60.

Updated Given that you've now asked a more advanced question, I'll extend my answer accordingly.

In general, $$s=np$$

There is as assumption on independence in that the occurrence or not of one success has no influence on the probability of subsequent successes.

At a more sophisticated level, the 60 Heads is termed the Expected Value, $E(x)$. It's the number of Heads that, on average, you would expect if you performed the experiment many times but, of course, it's perfectly possible to get more or less than the 60 Heads.

The standard deviation measures the amount of this variation in the likely number of heads. For our example $\sqrt{VAR(x)}$, of the number of Heads is 6.48. This very approximately means that most of the times this biased coin is tossed 200 times you'll get between 47 and 73 Heads.

In general for a Binomial Distribution, $$E(x)=np$$ $$VAR(x)=np(1-p)$$ and most results, $R$, lie between approximately, $$E(x)-2\times \sqrt{VAR(x)} \lt R \lt E(x)+2 \times \sqrt{VAR(x)}$$ This later approximation becomes less accurate as the probability drifts away from 0.5 and the distribution becomes skewed.

Back with our example the binomial probability of getting exactly 60 heads is given by, $$R_{60}= {200 \choose 60} \times 0.3^{46} \times 0.7^{154}$$ In general, $$R_{r}= {n \choose r} \times p^{r} \times (1-p)^{n-r}$$

For sample sizes of above 50 the interest in a specific probability such as $R_{60}$ lessens, (because it tends to zero as $n$ tends to infinity) and it's more meaningful to consider Cumulative Binomial Probabilities.

For these, tables can be used although, even in schools, the drift is now firmly towards using the excellent and inexpensive calculators that are now available such as the Casio ClassWix fx991ex for £22 : https://www.amazon.co.uk/Casio-FX-991EX-S-UH-Scientific-Calculator-Resolution/dp/B0719FWP3X

The Edexcel exam board, for example, no longer provide Normal Distribution Tables in their mathematics examinations; you have to use a calculator. The Binomial Cumulative Distribution the tables are still provided for a few values of sample size and probability of success and questions are asked where it's far easier to use the tables, as well as questions where it's far easier to use the BINOMIAL CD function on the calculator.

Moving up a level again, provided the probability of success is not to adrift of 0.5, and the sample size is large, say over 50, the Binomial Distribution can be modelled by a Normal Distribution.

However, unless indicated otherwise, I'll assume this is beyond what your question is asking.