# Question from Binomial probability.

As an analyst, I observe that DJIA fulfilled my client’s expectations (defined as “success”) three times out of past 5 yrs and underperfomed two times. What is the probability of exactly two sucesses in the next 5 years?

Doubt: How can we calc this? Wouldn’t we be assuming that past yr returns guide future forecasts, which I believe violates the assumptions about Binomial RV in the first place. My take is that more info is required to solve this. Any advice? Anyone?

Evidently, they want you to calculate P(success) = 3/5: you had three successes in five tries (historically). Use this in the binomial formula.

(By the way, that estimate of P(success) is known as the maximum-likelihood estimate. There are other estimates that you could use. One example is the Bayesian estimate, which in this case would be 4/7 (for k successes in n tries, the Bayesian estimate is (k + 1) / (n + 2)). Interestingly, I independently developed the Bayesian estimate for P(success) when I was working on an algorithm to predict how to bet against the spread in football games; it was years later that I was reading a statistics textbook and learned that I’d developed a real formula. You don’t need to know any of this parenthetical stuff for the Level I CFA exam.)

How is it that I do not violate the assumption of not using historical returns to predict future and still use historical returns to calculate “success”? Exactly two successes in next two yrs would use binomial distribution formula. Can we use it if we use historical data?

You have to estimate P(success) somehow. You’re not using historical data to predict success, only to predict the probability of success (we don’t distinguish between exceeding the client’s expectations by 1% or by 20%).

Use P(success) = 3/5 in the binomial formula. That’s what they want you to do.

Yea. So you are using historical data to predict the probability of future. How does that not violate the assumption? This brings me to my next question. Haven’t I understood the assumptions clearly? Can yiu explain the assumptions?

What assumption do you believe has been violated?

That past yr data cannot be relied on to predict future data, but wait isn’t that how probability is defined in the first place?

Um . . . yup. (At least for empirical probability.)

Then why is it mentioned…umm never mind…could you tell me the assumptions I need to have in mind while calculating binomial probability? (A loosely put term here)

The primary assumptions are:

• The Bernoulli trials are independent: knowledge of success on one trial doesn’t give you any information about success on another trial (i.e., the conditional probability of success on trial n given success on trial m is the same as the (unconditional) probability of success on trial n)
• The probability of success is constant: P(success) on trial n = P(success) on trial m

Yea. So in my case assumption 1 is violated. We are basing future prob I.e. p (success) on the past prob. If they were independent we we wouldn’t conclude anything about P (success) What am I missing here?

We’re not basing future probability on past probability; we’re estimating the probability based on the evidence.

We’re not saying that because we had a success on trial 47 it’s more likely we’ll have a success on trial 48; we’re saying that based on trials 1 - 47 it appears that the probability of success is 29/47 (= 0.617, maximal liklihood), or 30/49 (=0.612, Bayesian). We’re assuming that that probability is constant; that the success on trial 47 itself has no influence on the success on trial 48.

okay. Thanks.