Let’s work through each part of this question.

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Part (a)

1. Percentage of times heads came up in the first 100 tosses:

   $\text{Percentage of heads} = \left(\frac{36}{100}\right) \times 100 = 36\%$

   Answer 1 Question 5: 36%

2. Net gain or loss after 100 tosses:

   Since you win $1 for each head and lose $1 for each tail:

   $\text{Net result} = (36 \times 1) + (64 \times -1) = 36 - 64 = -28$

   This means you have a net loss of $28.

   Answer 2 Question 5: "a loss"

   Answer 3 Question 5: 28

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Part (b)

3. Checking consistency with the law of large numbers after 300 tosses:

   After 300 tosses, heads has come up 37% of the time, meaning:

   $\text{Number of heads} = 0.37 \times 300 = 111$

   The law of large numbers states that, as the number of trials increases, the proportion of heads should get closer to 50%. Since 37% is not close to 50%, this may not seem consistent with the law. However, statistically, fluctuations are still possible, even with a larger number of trials.

   Correct answer: B

   Explanation: The change is consistent with the law of large numbers because, as the number of trials increases, the proportion should approach 50%, though it may take many trials to converge.

4. Net gain or loss after 300 tosses:

   With 111 heads and 189 tails:

   $\text{Net result} = (111 \times 1) + (189 \times -1) = 111 - 189 = -78$

   So, you have a net loss of $78.

   Answer 4 Question 5: "a loss"

   Answer 5 Question 5: 78

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Part (c)

5. Number of heads needed to break even after 400 tosses:

   To break even after 400 tosses, you need to end with equal numbers of heads and tails. Since 400 tosses divided equally means 200 heads and 200 tails:

   • Currently, you have 111 heads from the first 300 tosses.
   • So, you need $200 - 111 = 89$ heads in the next 100 tosses to break even.

   Answer 6 Question 5: 89

   This is unlikely because getting 89 heads out of 100 tosses is an extreme outcome (89% heads), which is far from the expected 50%.

   Answer 7 Question 5: unlikely

   Answer 8 Question 5: far

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Part (d)

6. Explanation of the gambler's fallacy:

   Correct answer: D

   Explanation: This belief illustrates the gambler's fallacy because the probability of getting heads remains constant at 0.5 with each toss, regardless of past outcomes. Past losses do not increase the probability of winning in future tosses.

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Summary of Answers:

1. Answer 1 Question 5: 36%
2. Answer 2 Question 5: "a loss"
3. Answer 3 Question 5: 28
4. Answer 4 Question 5: "a loss"
5. Answer 5 Question 5: 78
6. Answer 6 Question 5: 89
7. Answer 7 Question 5: unlikely
8. Answer 8 Question 5: far

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Would you like more detailed explanations on any part?

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Here are five follow-up questions for further exploration:

1. How does the law of large numbers apply in coin tosses over a very large number of trials?
2. What is the probability of getting exactly 50 heads out of 100 coin tosses?
3. How does the gambler’s fallacy affect decision-making in real-life scenarios?
4. What are the chances of having a net gain after 300 tosses if the coin is fair?
5. How could understanding probability and the law of large numbers improve gambling behavior?

Tip: Remember, each coin toss is independent, and past results do not influence future probabilities.