Probability of $E$ when $F$ has happened - three coins tossed

Question

In conditional probability example where 3 fair coins are tossed, the sample space is:

HHH, HTH, HHT, THH, THT, TTH, TTT, HTT

$E$ be an event where at least two heads appear, $F$ be an event where the first coin yields a Tail. Now $P(E|F)$ warrants the cases satisfying event $F$ to act as a sample space for event $E$.

It is worded "probability of $E$ when $F$ has happened" - if $F$ has happened then $E$ can also happen separately after event $F$ is over, then there won't be any need for altering the sample space. Why is it not like that?

Answer 1

Lets say that the event F has happened, that only means that on tossing the first coin we get a Tail, now the statement:

probability of E when F has happened

means that first coin got tail, now what is the probability of event E i.e. getting two heads?

Now if you got tail in the first go, then you are left with only two coins to toss, the sample space has now been altered & that would be a decisive factor in calculating probability for event E.

Sample space for F = THH, THT, TTH, TTT -> now only in once case the event E is satisfied, hence the probability is 1/4.

Answer 2

P(E|F) is a conditional probability or the event that Event E happens given Event F. This is the same as $\frac{|E \cap F|}{|F|}$

There are $2^2$ outcomes for Event F. THH is the only outcome for at least two heads given that the first flip is T. You would get $\frac{1}{4}$

Note: I'm pretty sure this is different than P(E $\cap$ F) which would be $\frac{|E \cap F|}{|S|}$, where S is the sample space.

Answer 3

It is worded "probability of E when F has happened" - if F has happened then E can also happen separately after event F is over, then there won't be any need for altering the sample space. Why is it not like that?

You are providing a reasonable interpretation of the sentence in English, which does not represent the intent of the author.

Probability of E when F has happened is actually intended to have one of the following two meanings, neither of which is consistent with your interpretation:

• Probability that E has occurred given that F has occurred. This means that it is presumed that the trial representing whether the event E occurred or not took place at the exact same time that the event F occurred.

• Probability that when F occurs (at some point in the future) when the trial of event E simultaneously occurs, event E will succeed.