# A fair coin flip, which of the events are independent

how can i do this problem?

A fair coin if flipped 3 times. If (F1, F2, F3) denotes a typical flip sequence, let E1 denote the event that at least two of the Fi's are Heads, let E2 denote the event that exactly two of the Fi's are Heads, and let E3 denote the event that all the Fi are the same. Which of the pairs of these three events are independent?

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Next time, please post what you have tried. =) – TakeS Feb 21 '13 at 2:06
don't even know how to start the problem – heytherehowareyou Feb 21 '13 at 2:08
Sure you do. Either you know the definition of independent, in which case you can start writing down equations, or you don't, in which case you know the way to start the problem is by looking up the definitions of the words that are in the problem. – Gerry Myerson Feb 21 '13 at 2:20

We have $\mathbb P[E_1]=1/2, \mathbb P[E_2]= 3/8, \mathbb P[E_3] = 1/4.$ The only two events that are independent are $E_1$ and $E_3$, as $$\mathbb P[E_1 \cap E_3] = \mathbb P[\text{all heads}] = 1/8 = \mathbb P[E_1] P[E_3].$$