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Math Help - Growth rate and regular coins, and one unfair coin.

  1. #1
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    Growth rate and regular coins, and one unfair coin.

    I'm completely lost on this one; I haven't encountered a question like this before (with two growth rates).

    So, there's this experimenter who tosses 4 coins (this is toss 1) and about half of them (2) land with heads up. She adds the amount that landed heads up (2) to the intial number of coins (4) for a total of 6 coins. Then she tosses these 6 coins, and about half of them (3) turn up heads. She adds the 3 to the 6 to start the next (third) toss with 9 coins. This pattern of about 50% of the coins turning up heads continues, and we can look at this as exponential growth, with a growth rate of 50% or 0.5. An exponential equation for this experiment would be:
    C=4(1+0.5)^n (where C is the total number of coins at toss n). Ok, I get this so far and came up with the equation myself.

    Here's the confusing aspect:
    "The experimenter claims that the equation is explained by the formula for compound interest: A=P(1+i)^n. She argues that P represents the number of coins she started with, and i is 0.5 since the growth rate is about 50% (since about one-half of the coins tossed come up heads), and n is the number of tosses, which is like the compounding period. If her hypothesis is correct, create a formula that predicts the total number of coins if an unfair (weighted) coin is used that only comes up heads 1 out of every 4 times."

    So here's what I have:
    The probability of the unfair coin turning up heads is 25%.
    The probability of a regular coin turning up heads is 50%.

    What are the next logical steps? How would I go about answering this question? I'm not sure if I even understand it all that well. I'd really appreciate any help. I'm lost. :-(
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Growth rate and regular coins, and one unfair coin.

    I think you may be misreading the question, but it's so poorly worded it's hard to say. Where it says "create a formula that predicts the total number of coins if an unfair (weighted) coin is used that only comes up heads 1 out of every 4 times" I think what they really mean is "create a formula that predicts the total number of coins if unfair (weighted) coins are used that only come up heads 1 out of every 4 times." In other words ALL the coins are unfair, not just one. I think that if the meant that only one of the original 4 coins is unfair they would have written this as "create a formula that predicts the total number of coins if one unfair (weighted) coin is used among the first 4 that only comes up heads 1 out of every 4 times."
    Last edited by ebaines; January 11th 2013 at 01:09 PM.
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    Re: Growth rate and regular coins, and one unfair coin.

    Thanks for your input!

    Wouldn't the question be too easy if all the coins are unfair? The formula would then be: A=4(1+0.25)^n where A is the total number of coins, the growth rate is about 25. Seems too straight forward, unless I'm wrong.

    On the other hand, I have no idea how to answer if both fair and one unfair coins were used.
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    MHF Contributor ebaines's Avatar
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    Re: Growth rate and regular coins, and one unfair coin.

    Also note that they don't explain the rules for adding coins - are they to be fair coins or unfair coins? Or are the coins mixed so the some are fair and some are unfair? Does it depend on whether the coins that produce the heads are of the fair or unfair varierty? Whether specifying these rules you can't answer the question, which again causes ne to think that it's assumed all coins are of the unfair variety.
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    Re: Growth rate and regular coins, and one unfair coin.

    You might be right that all coins are unfair, then. This question was part of a lesson concerning Transformations on Quadratics, Transformations on Exponential Functions, and Transformations on Logarithmic Functions. We learned about compound interest, bacterial growth, and radioactive decay in the previous lessons. I think if read the question as it is written (thinking all but one coin are fair coins), then I think this question would be too advanced for this unit... Am I correct in thinking that? Or should I have the knowledge to figure that out given the subjects taught?
    Last edited by greatapt; January 11th 2013 at 01:29 PM.
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  6. #6
    MHF Contributor ebaines's Avatar
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    Re: Growth rate and regular coins, and one unfair coin.

    Yuo could probably figure it out of they told you what the rules are - how many of teh coins added are unfair. For examnple if every 4th coin added is unfair, then as the number of coins gets large the average return is 3/4 x 0.5 + 1/4 x 0.25 = 0.4375.
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