Sample Space of rolling a die and flipping 2 coins
By Sophia Bowman
Having some trouble understanding the sample space of rolling a die and flipping 2 coins So I assume the sample space of the above is 24 ($2^2$*$6^1$) ie: S = {1HH, 1HT, 1TH, 1TT, ...... 6TT}
But isn't the events 1HT and 1TH the same thing? Since we don't count 1HH and HH1 as two different events why would 1HT and 1TH be 2 separate events?
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$\begingroup$For now, disregard the rolling the die part. Only focus on $HT$ and $TH.$ Think of flipping two coins. By your logic, if $HT$ and $TH$ are the same thing then the probability of rolling $HH$ is $\frac{1}{3},$ $HT/TH$ is $\frac{1}{3},$ and $TT$ is $\frac{1}{3}.$ But of course, this is wrong. To put this into perspective, imagine flipping $1000$ coins. The probability that all $1000$ are heads is obviously not the probability that only $500$ are heads.
In other words, flipping $HT$ has a $\frac{1}{4}$ chance. Flipping $TH$ has a $\frac{1}{4}$ chance. But these probabilities don't overlap each other. You don't get $HT$ and $TH$ at the same time. If they were the same thing, then the probabilities would overlap.
I remember this was a difficult concept for me to grasp too as a novice. I like to think about putting things on a huge scale - instead of flipping $2$ coins, we flip $10000.$
$\endgroup$ 1 $\begingroup$Since we don't count 1HH and HH1 as two different events why would 1HT and 1TH be 2 separate events?
It is to be assumed that there are three events that occurred in some order:
- Event E_1 : 1st coin flip
- Event E_2 : 2nd coin flip
- Event E_3 : rolling the die.
When you blur 1HH and HH1, you are merely saying that the roll of the die can be considered the first event or the third.
However, when you blur 1HT and 1TH, you are saying that Heads on the first coin flip, Tails on the second coin flip represent the same combination of events as Tails on the first coin flip, Heads on the second coin flip.
This coin-flip-event blurring leads to trouble. For one thing, it confuses that the chance of exactly 1 Heads and 1 Tails (in some order) on the coin flips is (for example) twice the chance of two Heads on the two coin flips.
Further, many probability problems will focus on (for example) something like flipping three Heads in a row. In such a problem, maintaining a rigorous order that the coin flips occurred in will be critical.
In summary, blurring when the die roll occurred in relation to the coin flips is harmless in this instance, because there is only one die roll (event) in your example.
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