You flip a coin two times. The events A="two tails" and B="heads on the first flip" are

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BlackTom AI · Accepted Answer

Question recap: We’re flipping a fair coin twice, and we’re comparing two events A and B. A is 'two tails' and B is 'heads on the first flip'. 
Option 1: mutually exclusive. This would mean A and B cannot both occur in the same experiment. Since A requires the first flip to be tails (and the second tail as well), while B requires the first flip to be heads, the two events cannot happen together in a single run. Therefore, they do not occur simultaneously, which is the definition of mutually exclusive in probability.
Option 2: not mutually exclusive. This would claim that there exists at least one outcome where both A and B occur at the same time. However, no single outcome satisfies both conditions: A demands first flip = tails and second flip = tails, while B demands first flip = heads. These demands contradict each other for the first flip, so there is no overlap between A and B. As a result, this option is incorrect.
Additional clarity: When considering two events, looking at their intersection helps determine mutual exclusivity. If the intersection is empty (A ∩ B = ∅), the events are mutually exclusive. Here, A ∩ B = ∅ because no outcome has first flip both heads and tails simultaneously. A helpful way to visualize is listing all possible outcomes of two coin flips: HH, HT, TH, TT. A occurs only in TT, while B occurs in HH and HT; there is no common outcome, reinforcing their mutual exclusivity.
In summary, the reasoning shows that A and B cannot occur together, which aligns with the concept of mutual exclusivity in probability.

You flip a coin two times. The events A="two tails" and B="heads on the first flip" are

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