How do I distinguish that this question requires me to use total probability or the approach to take?

Question

Consider the simplified problem.

Pick an integer from 1-3 randomly. Whatever # is picked, then flip that many coins, i.e., if the integer picked, say, n=2, then we flip two coins. What is the probability that every coin flipped lands on heads?

When considering the question, I first thought of the following approach.

Idea 1: Let E be the event that all coins flipped lands on heads.

For $n=1$, $P(E)=\dfrac{1}{2}$; $n=2$, $P(E)=\dfrac{1}{2^2}$; and $n=3$, $P(E)=\dfrac{1}{2^3}$. So, my answer would be $P(E)=\dfrac{1}{2^n}$.

But a hint on the question, says, to use total probability. In that case, my answer would be

$P(E)=\sum\limits_{k=1}^3P(E|n=k)P(n=k)=\dfrac{1}{3}\left(\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}\right).$

My question is what in the question clearly points out which approach should I take. My thought is that the second approach needs to be taken because event $E$ results from an outcome of numerous possible events. (Second Question) If this is the correct approach, then how would the question be stated so that the first approach is correct?

Answer 1

The mistake on the first approach is in the notation you use. You said $$\text{Pr}(E)=\frac{1}{2^n}\ \ \text{instead of}\ \ \text{Pr}(E|n)=\frac{1}{2^n}$$ $\text{Pr}(E)$ is an unconditional probability, $\text{Pr}(E|n)$ is the conditional probability of $E$ given the number of coins tossed is $n$. And to compute $\text{Pr}(E)$ we need to consider all possibilities with an average weight given by the probabilities for each case for $n$: $$\text{Pr}(E)=\sum_{n\in \{1,2,3\}}\text{Pr}(E|n)\text{Pr}(n)$$ thats what the Law of Total Probabilities does.

Answer 2

Your posted (Math-education) question actually involves a deeper (Math-education) question, that goes well beyond the specific Math problem that your posting is focusing on. In my opinion, it is impossible to effectively learn any specific topic in Math, unless you treat that topic as a foreign language.

This means that you have to develop intuition/experience so that you naturally think in terms of the foreign language.

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As an introductory clarifying example:

Imagine that you do not speak Portugese, but that you are going to be spending some time in Portugal. Further imagine that everyone that you are exposed to, in Portugal, only speaks Portugese.

You have two choices for communication methods:

• You can bring Portugese-to-English and English-To-Portugese translation books. Then, you think of a sentence in English, and then translate it into Portugese, one word at a time.
  
  Then, you list to the responder's Portugese sentence. You translate that sentence back into English, one word at a time.
  
  This approach, which I refer to as thinking in English, makes sense if you are only going to be in Portugal for a few days.

• The alternative approach is to follow the exact same educational route that $4$ year old Portugese children follow. As they are taught letters of the Portugese alphabet, words, and sentences, they are always thinking in Portugese, so there is no translation-process involved.

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The analogy in the study of Math is that for (virtually) any Math, below the graduate-school level, you can find a pertinent Math textbook, on the topic, that is right-for-you. A Math textbook that is right for someone else might not be the textbook that is right for you.

In my opinion, $75\% - 80\%$ of the learning (AKA learning to think in the foreign language called Math) takes place in the exercises. So, you will want the book to contain many, many exercises for you to attack.

Analogizing to the $4$ year old Portugese child, suppose that you find the right Math textbook, open it to page 1, work all the way through to the end, and do not skip any of the exercises. The result will be that, with respect to this topic, you think in Mathematics.

Typically, with such a Math textbook, before a specific exercise is presented, the author will present Theorems, worked examples, and previous exercises that lead up to the specific exercise. Regarding these examples/theorems/previous-exercises as tools, it is important that each tool be crystal-clear to you.

Naturally, there will be places in the Math textbook that are relatively poorly written, or in error. So, in order for you to learn, you have to seek outside help to perfectly resolve each situation (i.e. the scorched-earth approach).

Then, similar to the specific Math problem that was posted, you may sometimes be unsure how to use the tools. For example, you might find two different ways of using the tools, which lead to two different answers.

This is a symptom that you have not yet mastered how the tools are to be used. Here, I advise that you adopt both of the following remedies:

• Attempt to independently determine which answer is accurate. This attempt may involve your taking a more basic manual computation approach, to see what the right answer is. In (for example) Combinatorics problems, this may involve writing a Computer program (i.e. in Python, C, or Java) to simulate the situation.
  
  An alternative method of independent determination is to do exactly what you did. Post the question on a forum, like MathSE, present two distinct methods of attack, each of which seem plausible, and ask for help determining which method (if any) is accurate/valid.

• In addition to the previous bullet-point, I also recommend reviewing the tools. The idea is that you (perhaps) did not master the underlying concepts in the tools leading up to the problem. This caused you to be confused as to how to apply the tools.
  
  This type of work-flow is bad, if your goal is to learn to (fluently) think in terms of this Math subject, as a foreign language.
  
  By analogy, this is similar to the Portugese child who has trouble learning how to spell words.