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So I have written a solution to a simple problem and it is as follows: Let $E_{1}$ denote dealer gets a black jack and $E_{2}$ denote that the player get a blackjack. Note that there are $\binom{52}{2,2,48}$ outcomes in total, and \begin{align*} P\left(E_{1}\right) & =P\left(E_{2}\right)=\frac{\dbinom{4}{1}\dbinom{16}{1}\dbinom{50}{2}}{\dbinom{52}{2,2,48}},P\left(E_{1}\cap E_{2}\right)=\frac{\dbinom{4}{1}\dbinom{16}{1}\dbinom{3}{1}\dbinom{15}{1}}{\dbinom{52}{2,2,48}},\\ \implies & P\left(E_{1}\cup E_{2}\right)=P\left(E_{1}\right)+P\left(E_{2}\right)-P\left(E_{1}\cap E_{2}\right)=0.095,\\ \implies & P\left(\text{neither gets blackjack }\right)=P\left(\left(E_{1}\cup E_{2}\right)^{c}\right)=1-P\left(E_{1}\cup E_{2}\right)=0.905. \end{align*}

My question is that is it legit to use $\implies$ in the middle of calculation? I don't usually see style like this often so just want to make sure..

Mathemagician
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2 Answers2

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It certainly happens sometimes that people write chains of computations this way, but it's in the context of a longer chain and in situation where there is basically nothing to say in between steps. In this case it would probably be more readable if you separated it out into three lines of computation with exposition in between. Even something as little as "thus" or "therefore" in between the lines would work.

Jared Holshouser
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The use of $\Longrightarrow$ at the start of a new line to indicate an equation follows from another adds very little meaning to the text, so I would avoid it. Such writing clearly alludes to the use of some statement of the form $$A\Longrightarrow B.$$ That is, it is using some implication, where we already know $A$. That is, if you use this notation, the fact that $B$ follows from $A$ should be very clear. This is most appropriate for long symbolic computations, but even then, it's unnecessary.

However, as the argument is set, each line is not really a consequence of the previous line - each line is introducing some new material. For instance, the fact that $$P(E_1\cup E_2)=P(E_1)+P(E_2)-P(E_1\cap E_2)$$ does not follow from the previous line of your proof, so it feels odd to see the implication written as if that were the case. The next line is similar. Thus, even the meaning that $\Longrightarrow$ does have is not really being served by the proof.

In general, the proof would be just as clear if you simply removed the implication symbols - regardless of whether they're there, the reader is going to assume each statement in a proof is somehow justified by axioms and previous statements - if anything needs to be clarified, it is what justifies the statement, not that it is justified. It would be even better to describe in words what each step is accomplishing, framing the equations between text, since that can help the reader understand the intention behind a proof.

Milo Brandt
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