Binomial Distribution

Binomial is one of the most applicable & relevant distributions to the real world events.

This type of distribution is applicable to instants that the outcome of an event could only express itself in two ways: Yes or No, Head or Tails, Ture or False,….

Let’s dissect the topic via a real world example.

Say you came across to 3 possible options, as vendors, to buy vacuum cleaner in an online platform, such as www.digikala.com.

Let’s assume you fell in the category of people who spend excessive time on reading reviews and doing researches for buying a brand that eventually does not differ much from the others.

Back to your experience of online commodity purchase; the 3 choices look like as following:

1.????? 10 Reviews – 100%

2.????? 50 Reviews – 96%

3.????? 400 Reviews – 93%

One common intuition usually dictates that larger the data, higher the confidence.

But there is one simple solution here,

In the first scenario, if you add two events (One positive and the other negative review) to the total events and make it 12. So, if we pretend that the positive reviews are 11 out of 12, it give the value of 91.7%, which is the probability of you having a good experience with this vendor.

Meanwhile, how can you trade off the greater amount of data or records, with the lower rate?

Same outline applies to the two other sellers.

• 49/52 = 94.2%
• 373/402 = 92.7%

It seems the best option is the second one, and does not have as much data (reviews) as the third one!

So, to understand the concept which in essence is the Laplace rule of succession, continue to read please.

Let’s assume sellers produce negative and positive experiences randomly, BUT each vendor has an underlying constant of 75% of giving a positive experience.

?P(Positive) = 75%,

75% = S (Success Rate)

You can guess the issue now; we are not aware of the underlying success rates of the 3 sellers of the vacuum cleaners.

So, given that we don’t know the success rate, we tend to maximize our chance of having a positive experience by selecting the proper seller/ brand.

Probability of Probabilities

If we look closely, we find out that there is countless possibilities of success rate between 0 to 1, for each seller. If we imagine the graph of these success rates plotted against the likelihood of each to occur, the outcome is somehow probability of probabilities ??.

This is unlike the coin flips and rolling dices, because the explained case actually happens in real life.

So if we repeat the events n times, for the second seller for instance, with 95% underlying succession rate and count the events in which 48 out 50 (96%) occurred, we can plot an histogram (the number of occurrences lies between certain intervals).

The formula for this case is:

To figure which vendor to choose, we still have to learn about probability density function, and Bayesian theorem, to be addressed in upcoming chapters.