Coin Toss experiment. Need Help

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Matt2015

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Hello, I am stuck on a question in one of my advanced functions courses and I could use some help. This question requires me to run an experiment of tossing coins into the air(starting with 20) and then removing every coin that lands on heads and recording the number of coins left, then I record the number of coins left and repeat this until I am at 0 coins. After generating my results I need to answer some questions that I am not fully understanding.


Toss | Coins remaining.

0 20
1 9
2 4
3 3
4 1
5 1
6 0

1 a) Graph the result. No problem here.

b) State an equation that best fitst the results. Use n to represent the number of the toss.
The only equation I can think of relates to A=P(1+I)^n (same formula for compound interest.
P= # of coins you start with , N =6 for 6 tosses. 20(1+I)^6... I need help here.

c)What physical Phenomenon does this model?

I am confused here aswell. Do they refer to the equation or the graph. The graph looks like a Logarithmic function with Y axis as an asymptote. The equation looks like the formula for compound interest.

D) State the equation that would best predict the number of coins if you began with N_0 coins.

This looks like the same N_0 used in an equation for elemental decay. N(t)=N_02^(-t/d) but this doesn't fit into this information so I keep thinking back to N-o(1+i)^n.
20(1+i)^n

any chance anyone can give me some insight on these questions? Just even a boost in the right direction. I am quite confused.
 
(b) Yes, except you have it slightly wrong depending on just what you are referring to. Since the y value of the graph (how many heads are left) is decreasing, it would be y(n) = P/(1+i)n where P is the amount you started with and n is the number of tosses. So yes, P = y (1+i)n would be the graph for compound interest, but this problem is in reverse, i.e. how much do you need to start with (what is y) to get a certain amount (P) in n years. So this looks like what is called a present value formula.

(c) As a general question they mean the equation [from part(a)] although if you performed the experiment, you might get results which looked like the graph.

d) Now this one could sort of bug me. Have you studied (linear) regression fits to data? If not, then the answer is a guess. One way to guess though is to consider just what you have done. Since you are tossing coins, how many heads would you expect to get if you tossed a lot of them. If they were fair coins, you would expect about half of them would be heads. So how many would you have left? And how many would you have left after you tossed those? And then those? And... Or, if you have studied regression, you could do an exponential fit to the data [that is take the logs of y (dropping the zero value) and do a linear regression]. Oh, and yes, it is the half life formula.
 
I have not studied regression so feel same to assume that this is not the answer for part D. What I have studied is exponential equations such as half life, compound interest and even logarithmic equations such as earthquake magnitude and sound.

For part b) you say y=(how many heads are left). In this question its counting the # of coins left after removing the heads. So its how many coins are left. I think it leads to the same answer. I figured this equation didn't make sense since I am removing coins not adding them. So that would lead to N=P / (1+i)^N or 6=20 / (1+i)^6. Do you think this is correct?

c) Still don't understand what they are asking. I want to go with the equation generated in part b) N=P/(1+i)^N , so the answer would be the equation looks like the physical phenomena of interest, or present value.
 
Matt2015 said:
I have not studied regression so feel same to assume that this is not the answer for part D. What I have studied is exponential equations such as half life, compound interest and even logarithmic equations such as earthquake magnitude and sound.

For part b) you say y=(how many heads are left). In this question its counting the # of coins left after removing the heads. So its how many coins are left. I think it leads to the same answer. I figured this equation didn't make sense since I am removing coins not adding them. So that would lead to N=P / (1+i)^N or 6=20 / (1+i)^6. Do you think this is correct?

c) Still don't understand what they are asking. I want to go with the equation generated in part b) N=P/(1+i)^N , so the answer would be the equation looks like the physical phenomena of interest, or present value.
Click to expand...

Sorry, yes you are right. It is supposed to be how many coins are left. So for (b) we have
(Eq. 1) y = P / (1+i)n
Since i is just some constant, let's let
a = ln2(1 + i).
Now, lets assume we did the experiment by tossing the coins which were left every minute. That way we can change n to t where t is measured in minutes. Thus we have
(Eq. 2) y = P 2-a t
Now, in one formulation, Equation (1), it looks like one of the present value formulas but in another formulation, Equation (2), it looks like a half life formula. So, from the behavior perspective, they are both the same.
 
Sadly this issue continues to elude me. However it is probably because I am trying to understand your responses while I am at work not 100% focused on the problem. Hopefully I can understand it better when I try it again at home.

One of you earlier responses you say y(n)=p\(1+I)^N. Recently you changed this to be Y=P\(1+i)^N. In the graph N(# of tosses) is graphed on the X axis while the dependent variable (Y) is graphed as the # of coins left after the toss. So does y(n) still apply or do I simply leave it as Y=20\(1+i)^6. None of these equations give me an answer when I try to plug in numbers for X to receive a Y variable so I am very lost still.
 
Matt2015 said:
Sadly this issue continues to elude me. However it is probably because I am trying to understand your responses while I am at work not 100% focused on the problem. Hopefully I can understand it better when I try it again at home.

One of you earlier responses you say y(n)=p\(1+I)^N. Recently you changed this to be Y=P\(1+i)^N. In the graph N(# of tosses) is graphed on the X axis while the dependent variable (Y) is graphed as the # of coins left after the toss. So does y(n) still apply or do I simply leave it as Y=20\(1+i)^6. None of these equations give me an answer when I try to plug in numbers for X to receive a Y variable so I am very lost still.
Click to expand...

I've given the form of the curve but not all of the values. We know, from the experiment, that P is 20 but we don't know what i (or, in the other formulation, a) is. That is really the question in (d). In fact we could do a regression fit to get a 'best fit' for P and i [they would be close to 20 and 1 respectively]. So, if P were 20 and i were 1, then a would be 1 and what formula(s) would you have?

EDIT: Notice that N is 6 only for the 6th toss. It is 5 for the 5th toss, 4 for 4th, ...
 
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