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Math Probability Coin Experiment

by TEN










































1. In your own words, describe two main differences between classical and empirical probabilities.
The first difference between the two is that classical probability is a theoretical computation whereas empirical probability is computed based on experiment or observation.
The second difference is that classical probability assumes the occurrence of any possible event within the sample space is just a likely as any other, where empirical probability makes no assumptions regarding possible outcomes.

2. Coin Experiment
20 coins
Random Trials (coins were put in a small plastic bag and shaken around)
Results:
1st trial: 10 heads; 10 tails
2nd trial: 11 heads; 9 tails
3rd trial: 10 heads; 10 tails
4th trial: 13 heads; 7 tails
5th trial: 12 heads; 8 tails
6th trial: 11 heads; 9 tails
7th trial: 8 heads; 12 tails
8th trial: 12 heads; 8 tails
9th trial: 10 heads; 10 tails
10th trial: 10 heads; 10 tails

P(E) = (number of times a specific event is observed) ÷ (total number of events observed)

Number of times a specific event is observed = number of heads observed, or number of tails observed

Total number of events observed = the total of heads and tails observed (this would also be equal to the total number of coins times the number of trials: 20 coins*10 trials = 200.)

Consider just your first count of the tossed coins. What is the observed probability of tossing a head? Of tossing a tail? Show the formula you used and reduce the answer to lowest terms.
First Count of the tossed coins:
Observed probability of tossing a head? P(E) = 10 heads/20 coins = ½
Observed probability of tossing a tail? P(E) = 10 tails/20 coins = ½

Did any of your ten repetitions come out to have exactly the same number of heads and tails? How many times did this happen?
Exactly the same number of heads and tails occurred in 4 trials (out of a total of 10 trials)

How come the answers to the step above are not exactly ½ and ½?
The two outcomes of a typical coin flip are not equally likely because of a bias. The fact that the coins were put in a container (or bag) and mixed up by shaking the container probably did remove the human bias.
However, the coins are inherently biased because the weight is not evenly distributed within the coin.
The coins probably end up with the heavier side down more often than not. Apparently, this means that heads-up appears more frequently.

Stanford University conducted a study of coin flips. You can download the results here: http://www-stat.stanford.edu/~susan/papers/headswithJ.pdf

What kind of probability are you using in this “bag of coins” experiment?
Empirical probability

Compute the average number of heads from the ten trials (add up the number of heads and divide it by 10).
Average Number of Heads = (10 + 11 + 10 + 13 + 12 + 11 + 8 + 12 + 10 + 10)heads/10 = 107/10 = 10.7

Change this to the average probability of tossing heads by putting the average number of heads in a fraction over the number of coins you used in your tosses.
Average probability of tossing heads = 10.7/20 = 0.535 = 53.5%

Did anything surprising or unexpected happen in your results for this experiment?
Yes. There was a natural bias in the results that was quite obvious.

3. Write the sample space for the outcomes of tossing three coins using H for heads and T for tails.
There are 8 possible outcomes:
H,H,H
H,H,T
H,T,H
H,T,T
T,H,H
T,H,T
T,T,H
T,T,T

What is the probability for each of the outcomes?
probability for each of the outcomes = 1/8

Which kind of probability are we using here?
Classical probability (theoretical computation).

How come we do not need to have three actual coins to compute the probabilities for these outcomes?
We do not need to have three actual coins because we assume the probability of each possible outcome is the same.

Comments for Math Probability Coin Experiment

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Mar 17, 2012
Math Probability Coin Experiment
by: Staff


Part I

Question:

by TEN


1. In your own words, describe two main differences between classical and empirical probabilities.

The first difference between the two is that classical probability is a theoretical computation whereas empirical probability is computed based on experiment or observation.

The second difference is that classical probability assumes the occurrence of any possible event within the sample space is just a likely as any other, where empirical probability makes no assumptions regarding possible outcomes.


2. Coin Experiment

20 coins

Random Trials (coins were put in a small plastic bag and shaken around)

Results:

1st trial: 10 heads; 10 tails
2nd trial: 11 heads; 9 tails
3rd trial: 10 heads; 10 tails
4th trial: 13 heads; 7 tails
5th trial: 12 heads; 8 tails
6th trial: 11 heads; 9 tails
7th trial: 8 heads; 12 tails
8th trial: 12 heads; 8 tails
9th trial: 10 heads; 10 tails
10th trial: 10 heads; 10 tails

P(E) = (number of times a specific event is observed) ÷ (total number of events observed)

Number of times a specific event is observed = number of heads observed, or number of tails observed

Total number of events observed = the total of heads and tails observed (this would also be equal to the total number of coins times the number of trials: 20 coins*10 trials = 200.)

Consider just your first count of the tossed coins.

What is the observed probability of tossing a head?


Of tossing a tail?



Show the formula you used and reduce the answer to lowest terms.

First Count of the tossed coins:

Observed probability of tossing a head? P(E) = 10 heads/20 coins = ½
Observed probability of tossing a tail? P(E) = 10 tails/20 coins = ½



Did any of your ten repetitions come out to have exactly the same number of heads and tails?


How many times did this happen?


Exactly the same number of heads and tails occurred in 4 trials (out of a total of 10 trials)


How come the answers to the step above are not exactly ½ and ½?

The two outcomes of a typical coin flip are not equally likely because of a bias.

The fact that the coins were put in a container (or bag) and mixed up by shaking the container probably did remove the human bias.
----------------------------------------------

Mar 17, 2012
Math Probability Coin Experiment
by: Staff

----------------------------------------------
Part II

However, the coins are inherently biased because the weight is not evenly distributed within the coin.

The coins probably end up with the heavier side down more often than not. Apparently, this means that heads-up appears more frequently.

Stanford University conducted a study of coin flips. You can download the results here: http://www-stat.stanford.edu/~susan/papers/headswithJ.pdf

What kind of probability are you using in this “bag of coins” experiment?

Empirical probability

Compute the average number of heads from the ten trials (add up the number of heads and divide it by 10).

Average Number of Heads = (10 + 11 + 10 + 13 + 12 + 11 + 8 + 12 + 10 + 10)heads/10 = 107/10 = 10.7

Change this to the average probability of tossing heads by putting the average number of heads in a fraction over the number of coins you used in your tosses.

Average probability of tossing heads = 10.7/20 = 0.535 = 53.5%

Did anything surprising or unexpected happen in your results for this experiment?

Yes. There was a natural bias in the results that was quite obvious.




3. Write the sample space for the outcomes of tossing three coins using H for heads and T for tails.

There are 8 possible outcomes:

H,H,H
H,H,T
H,T,H
H,T,T
T,H,H
T,H,T
T,T,H
T,T,T

What is the probability for each of the outcomes?

probability for each of the outcomes = 1/8

Which kind of probability are we using here?

Classical probability (theoretical computation).

How come we do not need to have three actual coins to compute the probabilities for these outcomes?

We do not need to have three actual coins because we assume the probability of each possible outcome is the same.

Answer:
You’ve done a good job at answering these questions.

The only changes I would make are shown below:


What is the observed probability of tossing a head?

107/200 = 0.535

Of tossing a tail?

93/200 = 0.465


Show the formula you used and reduce the answer to lowest terms.

First Count of the tossed coins:

Observed probability of tossing a head? P(E) = 107 heads/200 trials = 0.535
Observed probability of tossing a tail? P(E) = 93 tails/200 trials = 0.465



Did any of your ten repetitions come out to have exactly the same number of heads and tails?

Yes





Thanks for writing.

Staff
www.solving-math-problems.com



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