Question

suppose you play a coin toss game in which you win \\$1 if a head appears and lose \\$1 if a tail appears. in the first 100 coin tosses, heads comes up 34 times and tails comes up 66 times. answer parts (a) through (d) below.

a. the change is not consistent with the law of large numbers because, as the number of trials increases, the proportion should grow closer to 50\\%.
b. the change is not consistent with the law of large numbers, because the trials are not independent.
c. the change is consistent with the law of large numbers. because the percentage is low the first 100 trials, it has to be higher the next 200 trials to even out.
d. the change is consistent with the law of large numbers because, as the number of trials increases, the proportion should grow closer to 50\\%.

what is your net gain or loss at this point? select the correct choice and fill in the answer box to complete your choice.

a. you have lost \\$84. (type an integer.)
b. you have gained \\$. (type an integer.)

c. how many heads would you need in the next 100 tosses in order to break even after 400 tosses? is this likely to occur? select the correct choice and fill in the answer box to complete your choice.

a. you would need to toss heads. this is likely because it is close to the expected number of heads.
b. you would need to toss heads. this is unlikely as it is far from the expected number of heads.
c. you would need to toss heads. this is likely because so few heads have been tossed so far.

Explanation:

Identify the given parameters

We are given a coin toss game where:

• Winning a head gives \(+\$1\).
• Losing a tail gives \(-\$1\).
• In the first 100 tosses, heads came up 34 times and tails came up 66 times.
• The current net loss is \(34 \times 1 - 66 \times 1 = -\$32\).
• We want to find the number of heads needed in the next 100 tosses (making 200 tosses total) to break even after 400 tosses total.
• Wait, let's read part (c) carefully: "How many heads would you need in the next 100 tosses in order to break even after 400 tosses?"
• Let's re-read the setup. The image shows part of a multi-part question.
• Let's calculate the total number of tosses: 400 tosses.
• To break even after 400 tosses, the net gain must be \$0.
• This means we need exactly 200 heads and 200 tails out of the 400 total tosses.
• We already have 34 heads from the first 100 tosses.
• Let's check if there is an intermediate part we missed. The text says: "How many heads would you need in the next 100 tosses in order to break even after 400 tosses?"
• Wait, "the next 100 tosses" would bring the total to 200 tosses. But the question asks to break even after 400 tosses.
• Let's assume the next 100 tosses refers to a specific block, or perhaps the question implies that the remaining 300 tosses are split, or we assume the expected value for the other tosses.
• Let's re-read: "How many heads would you need in the next 100 tosses in order to break even after 400 tosses? Is this likely to occur?"
• If we need to break even after 400 tosses, we need 200 heads in total.
• We have 34 heads in the first 100 tosses.
• This leaves \(200 - 34 = 166\) heads needed in the remaining 300 tosses.
• If the "next 100 tosses" refers to the second block of 100 tosses, and we assume the remaining 200 tosses (tosses 201 to 400) result in their expected value of 50% heads (which is 100 heads):
• Heads needed in tosses 101 to 200: \(200 - 34 - 100 = 66\) heads.
• Let's verify this logic. If we get 66 heads in the next 100 tosses, and the remaining 200 tosses behave normally (getting 100 heads), the total heads will be \(34 + 66 + 100 = 200\), achieving a break-even state.
• Let's check the likelihood: getting 66 heads out of 100 tosses is highly unlikely for a fair coin, as the expected number is 50, and 66 is more than 3 standard deviations away (\(\sigma = \sqrt{100 \times 0.5 \times 0.5} = 5\)).

Calculate the required heads

• Total tosses: 400.
• Heads needed to break even: 200.
• Heads obtained in first 100 tosses: 34.
• Heads expected in the last 200 tosses: 100.
• Heads needed in the next 100 tosses: \(200 - 34 - 100 = 66\).

Determine the likelihood

• The expected number of heads in 100 tosses is 50.
• Getting 66 heads is far from the expected value of 50.
• Therefore, this is unlikely to occur.
• This matches Option B: "You would need to toss 66 heads. This is unlikely as it is far from the expected number of heads."

Answer:

• O A. You would need to toss 66 heads. This is likely because it is close to the expected number of heads.
• O B. You would need to toss 66 heads. This is unlikely as it is far from the expected number of heads. (Correct answer)
• O C. You would need to toss 66 heads. This is likely because so few heads have been tossed so far.