Could someone please help me solve the following question:
Say we have 3 coins which have probabilities 1/3, 1/2, 2/3 respectively of getting heads.
We choose a random order in which to flip the coins (all possible orderings being equally likely), once each. Let A_i = the event that the ith flip is heads. Are the events A_1 A_2 A_3 independent?
Attempt at solution: So the possibility of orderings for the coins are 123, 132, 213, 231, 312, 321 (if I label the three coins 1 2 and 3)
Do I then find A1, for example, by finding e.g. P(H first | 123)*1/6 for all the different combinations?
Is there a less tedious approach?
And then for checking independence, I would need P(A1 n A2), P(A2 n A3) etc, would I need to consider the same approach again e.g. P(A1 n A2) = P(HHT or HHH | 123)*1/6 + P(HHT or HHH | 132)*1/6 … ?
Please let me know if I’m on the right track or if I should take a different approach, thank you!