Sep 8, 2024

Coin Toss Possibilities in an expression

binomial theorem comes in handy

Okay, you've tossed a fair coin three times.
How do you usually represent the possibilities that might occur?

Something like this?

Yea that's fine, but, what if you can represent all this with just an algebraic expression?

Finding the expression #

Okay, let us allocate some variables.
Let's take $H$ for heads and $T$ for tails.

Possibilities:

• All Heads: $H^3$
  • the "cube" is because there are 3 heads $\text{HHH}$
• 2 Heads, 1 Tail: $3\times H^{2}T$
  • similar notation as the above for the 3 possibilities: $\text{HHT, HTH, THH}$
• 2 Tails, 1 Head: $3\times T^{2}H$
  • $\text{HTT, TTH, THT}$
• All tails: $T^3$
  • $\text{TTT}$

Now, you take all those terms together and you get:

And if you notice carefully, that is just the expansion for:

Pretty clever, eh?

Okay, what can you use it for? #

Suppose someone asks you what are the possibilities that you will get at least 2 heads with 3 coin tosses, you can just look into the expression:

And add up the possibilities: 1+3 = 4! Boom!

Yes, this comes especially handy in such easy permutation questions.

Wait a minute, isn't this the Binomial Expansion? YES IT IS! #

In fact, you can find the possibilities of any of such coin tosses using the :Binomial Theorem!

So, the possibilities of a fair coin tossed 69 times:

This method is used to represent the possibilities, not anything else.
So instead of writing all of the possibilities, just write the binomial expansion, man.

Note: this won't work for dices, because they have 6 faces, not a binomial.

What is new to this? #

Well, you just learned that you can find the possibilities of things usingexpressions.

Further Reading #

• Binomial Distribution (Probability)
• How To Understand Combinations Using Multiplication, BetterExplained

:x binomial #

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