Is flipping a coin 2 times the same as flipping 2 coins at the same time?

Question

My teacher asked us this question in class but no one could explain why flipping a coin 2 times in a row is not the same as flipping 2 coins at the same time.

Answer 1

When we talk about probability, we need to specify what the sample space and probability measure are.

If we flip the coin two times, then the sample space is $\Omega=\{HH,HT,TH,TT\}$ and the probability is the same for each point in $\Omega$.

However, if we consider flip two coins at the same time, the sample space becomes $\Omega=\{\{H,H\},\{H,T\},\{T,T\}\}$. (unordered pairs) Then $P(\{H,H\})=P(\{T,T\})=0.25$ bu t $P(\{H,T\}) =0.5$.

Maybe that's why they are different.

Answer 2

If the coins involved all have equal probability to produce e.g. a head then the probabilistic model of the first situation is also suitable for the second situation.

Answer 3

When it comes to combinatorics, they are not the same.

If you consider the case where you flip both coins at once, the set of events is $\Omega_1=\{HH, HT,TT\}=\{HH,TH,TT\}$ as there is no order (although one could define the meaning of order differently. In this case, the order is determined by when you toss the coin(s)). So the cardinality is $\vert \Omega_1 \vert=3.$

But if you consider the case where you flip the coins consecutively, the set of events is $\Omega_1=\{HH, HT,TH,TT\}$, thus cardinality is $\vert \Omega_2 \vert=4.$