(b) write the following event in set notation. the coin is tossed three times. choose the correct answer below. a. {hth, thh, htt, tht, tth, ttt} b. {hhh} c. {hhh, ttt} d. {hth, tthh, htth, htt, thtt, tht, ttth, tttt}

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# Brief Explanations:
To determine when the coin is tossed three times, we analyze the experiment: the coin is flipped until two heads appear, with a maximum of three flips. If two heads occur in less than three flips, the experiment stops. So, the coin is tossed three times only when we don't get two heads in the first two flips. Let's list the cases:
- First flip: T (tail). Second flip: T (tail) – then we flip a third time (outcomes: TTH, TTT).
- First flip: T, Second flip: H – but we stop at two flips (TH), so no third flip.
- First flip: H, Second flip: T – stop at two flips (HT), no third flip.
- First flip: H, Second flip: H – stop at two flips (HH), no third flip.

Wait, actually, the correct way is: the coin is tossed three times when the first two flips do not contain two heads. So the first two flips can be TH, HT, or TT. But if the first two flips are HH, we stop at two flips. If the first two flips are TH or HT, we still stop? Wait, no – the problem says "until two heads appear, up to a maximum of three flips. If two tails are flipped, the coin is still tossed a third time to complete the experiment." Wait, maybe I misread. Let's re-express the experiment:

- Flip 1: H. Then we need one more H. Flip 2: H – stop (HH, 2 flips). Flip 2: T – then we can flip a third time? Wait, no – the rule is "until two heads appear, up to a maximum of three flips. If two tails are flipped, the coin is still tossed a third time to complete the experiment." Wait, maybe the sample space for the number of flips:

Wait, the key is: the coin is tossed three times when the experiment doesn't end in two flips. When does the experiment end in two flips? When we get two heads (HH) or when we get two tails? No, the problem says "until two heads appear, up to a maximum of three flips. If two tails are flipped, the coin is still tossed a third time to complete the experiment." Wait, maybe the correct interpretation is: we flip until we get two heads, but we must flip at least three times? No, that doesn't make sense. Wait, the problem says "up to a maximum of three flips" – so we flip at most three times, stopping when we get two heads. But if we get two tails, we still flip a third time. Wait, maybe the sample space for the experiment (all possible sequences) is:

- Sequences with two heads in ≤2 flips: HH (2 flips), THH (3 flips? No, wait, if we get H on flip 1, T on flip 2, then we need another H, so flip 3: H (THH, 3 flips). Wait, no – the rule is "until two heads appear, up to a maximum of three flips". So:

- Flip 1: H. Flip 2: H – stop (HH, 2 flips).
- Flip 1: H. Flip 2: T. Then we can flip a third time (since max is three flips), so flip 3: H (THH, 3 flips) or T (THT, 3 flips). Wait, but we are looking for when the coin is tossed three times. So the coin is tossed three times when the experiment doesn't end in two flips. When does it end in two flips? Only when we get HH (two heads in two flips). If we get HT, TH, or TT in the first two flips, we have to flip a third time (because we haven't got two heads yet, and the max is three flips). Wait, no – if we get HT: first flip H, second flip T – we still need one more H, so we flip a third time. If we get TH: first T, second H – still need one more H, flip third time. If we get TT: first T, second T – we still flip a third time (as per the problem statement: "If two tails are flipped, the coin is still tossed a third time to complete the experiment"). If we get HH: we stop at two flips.

So the sequences where the coin is tossed three times are all sequences with length 3, i.e., sequences that didn't end at two flips (i.e., didn't get HH in two flips). So the possible three-flip sequences are:

- HTH: first H, second T, third H (we stopped at third flip because we got two Hs? Wait, no – HTH has two Hs (flips 1 and 3), so we would stop when we get the second H. Wait, maybe the experiment is: we flip until we get two heads, but we must flip at least three times? No, the problem says "up to a maximum of three flips" – so we flip at most three times, stopping when we get two heads. So:

- HH: stop at 2 flips (two heads).
- THH: stop at 3 flips (two heads at flip 3).
- HTH: stop at 3 flips (two heads at flip 3).
- THT: flip 1 T, flip 2 H, flip 3 T – only one H, so we flip three times (since we didn't get two heads, and max is three flips).
- HTT: flip 1 H, flip 2 T, flip 3 T – only one H, flip three times.
- TTH: flip 1 T, flip 2 T, flip 3 H – only one H, flip three times.
- TTT: flip 1 T, flip 2 T, flip 3 T – zero Hs, flip three times.

Wait, but the event "the coin is tossed three times" means that we didn't stop at two flips. When do we stop at two flips? Only when we get HH (two heads in two flips). So all other sequences (length 3) are when we tossed three times. Wait, but the options are:

A. {hth, thh, htt, tht, tth, ttt}

Wait, let's check each option:

Option A: hth (3 flips), thh (3 flips), htt (3 flips), tht (3 flips), tth (3 flips), ttt (3 flips). Let's see if these are the sequences where we tossed three times.

- HH: 2 flips (not in A)
- TH: 2 flips? No, TH would be two flips, but we need to get two heads, so TH has only one H, so we would flip a third time. Wait, I think my initial interpretation was wrong. Let's re-express the experiment properly:

The experiment is: flip the coin until two heads appear, but we will flip at most three times. So:

- If we get two heads in 2 flips (HH), we stop (2 flips).
- If we get two heads in 3 flips (THH, HTH), we stop (3 flips).
- If we don't get two heads in 3 flips (THT, HTT, TTH, TTT), we stop (3 flips).

So the sequences are:

- 2 flips: HH
- 3 flips: THH, HTH, THT, HTT, TTH, TTT

Now, the event "the coin is tossed three times" is all sequences with length 3, i.e., THH, HTH, THT, HTT, TTH, TTT. Wait, but option A is {hth, thh, htt, tht, tth, ttt} – which matches these (HTH, THH, HTT, THT, TTH, TTT). Let's check the other options:

Option B: {hhh} – no, HHH would have two heads at flip 2 (HH), so we would stop at flip 2, not flip 3. So B is wrong.

Option C: {hhh, ttt} – HHH is wrong (as above), TTT is in A, so C is wrong.

Option D: {thth, tthh, htth, httt, thtt, ttht, ttth, tttt} – these are four-flip sequences, but the max is three flips, so D is wrong.

Thus, the correct answer is A.

# Answer:
A. {hth, thh, htt, tht, tth, ttt}

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