  # Math - probability

In your own words, describe two main differences between classical and empirical probabilities.
Gather coins you find around your home or in your pocket or purse. You will need an even number of coins (any denomination) between 16 and 30. You do not need more than that. Put all of the coins in a small bag or container big enough to allow the coins to be shaken around. Shake the bag well and empty the coins onto a table. Tally up how many heads and tails are showing. Do ten repetitions of this experiment, and record your findings every time.
State how many coins you have and present your data in a table or chart.
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.
Did any of your ten repetitions come out to have exactly the same number of heads and tails? How many times did this happen?
How come the answers to the step above are not exactly ½ and ½?
What kind of probability are you using in this “bag of coins” experiment?
Compute the average number of heads from the ten trials (add up the number of heads and divide it by 10).
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.
Did anything surprising or unexpected happen in your results for this experiment?
Write the sample space for the outcomes of tossing three coins using H for heads and T for tails.
What is the probability for each of the outcomes?
Which kind of probability are we using here?
How come we do not need to have three actual coins to compute the probabilities for these outcomes?

### Comments for Math - probability

 Apr 13, 2011 Math – Coin Toss Probability by: Staff ------------------------------------------------------ Part II P(E) = (number of times an specific event is observed) ÷ (total number of events observed) Number of times an 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.) 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 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 Average probability of tossing heads 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. A natural bias in the results was obvious. 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. Thanks for writing. Staff www.solving-math-problems.com

 Apr 13, 2011 Math – Coin Toss Probability by: Staff Part I The question: In your own words, describe two main differences between classical and empirical probabilities. Gather coins you find around your home or in your pocket or purse. You will need an even number of coins (any denomination) between 16 and 30. You do not need more than that. Put all of the coins in a small bag or container big enough to allow the coins to be shaken around. Shake the bag well and empty the coins onto a table. Tally up how many heads and tails are showing. Do ten repetitions of this experiment, and record your findings every time. State how many coins you have and present your data in a table or chart. 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. Did any of your ten repetitions come out to have exactly the same number of heads and tails? How many times did this happen? How come the answers to the step above are not exactly ½ and ½? What kind of probability are you using in this “bag of coins” experiment? Compute the average number of heads from the ten trials (add up the number of heads and divide it by 10). 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. Did anything surprising or unexpected happen in your results for this experiment? Write the sample space for the outcomes of tossing three coins using H for heads and T for tails. What is the probability for each of the outcomes? Which kind of probability are we using here? How come we do not need to have three actual coins to compute the probabilities for these outcomes? The answer: 1. TWO MAIN DIFFERENCES between classical and empirical probabilities First Difference Classical probability is a theoretical computation. Empirical probability is computed based on experiment or observation. Second Difference Classical probability assumes the occurrence of any possible event within the sample space is just a likely as any other. Empirical probability makes no assumptions regarding possible outcomes. 2. COIN EXPERIMENT 20 coins Random Trials 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 ------------------------------------------------------

 Aug 04, 2011 numbers-probability by: Anonymous Numbers are the Supreme Court of science. However Godel proved that we may not prove everything. There are Physics Foibles!!

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