07 Jan “The Law of Large Numbers

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m4_coin_flip.jpg

Probability

This discussion activity will familiarize you with probability using an applet. The applet is a simulation of flipping a fair coin a given number of times and then estimating the probability of getting a head using the relative frequency approximation of probability.

The probability of getting a head or tail on one flip of a fair coin is 0.5 or 50%. The relative frequency approximation of probability says that the probability of an Event A is estimated by dividing the number of times that Event A occurred by the number of times the trial was repeated. For example, if you flip a coin 10 times and get 4 heads, then you could estimate the probability of getting a head as 4 divided by 10 or 0.4.

Run Applet & Reflect

Use the Simulating the probability of a head (with a fair coin) (Links to an external site.) applet and follow steps 1 through 5.

At the top left side of the applet, under “Coin Flipping”, select “5 flips”. After the simulation finishes the 5 flips, select “5 flips” again for a total of 10 flips of the coin. Record the number of heads (given in the box labeled Count) in 10 flips of a coin and the cumulative probability of a head based on the ten flips. (The cumulative probability is given in the box labeled Proportion.) Is the number of heads what you expected in 10 flips?

Now select 5 flips four more times (you will need to wait for each 5 flips to complete before clicking again.) You should now have 30 flips of the coin (see the Total column). Record the number of heads in 30 flips and the cumulative probability of a head based on 30 flips. What is the longest string of consecutive heads or tails that you got in the 30 flips? Do you think that is unusual? (In the Flips box you can use the scroll bar to look at the individual coins to find consecutive heads or tails. You can also select the box labeled “Convergence” to locate consecutive strings. When you flip heads the line rises and when you flip tails the line falls.)

Now select 1000 flips. This should give you 1030 flips of the coin. Record the number of heads and the cumulative probability of a head now.

Select 1000 flips one more time so that you have 2030 flips. Record the number of heads and the cumulative probability of a head now.

“The Law of Large Numbers states that if an experiment with a random outcome is repeated a large number of times, the empirical probability of an event is likely to be close to the true probability. The larger the number of repetitions, the closer together these probabilities are likely to be” (Text, p. 224). Does the coin flipping process you just completed illustrate the Law of Large Numbers? Why or why not?

Post

Share your results and reflections from the five steps of the applet activity.

Your initial post should be made before the end of the fourth day of the module to receive full credit. Return at least once later in the module to comment on two or more classmates’ posts. Do NOT “post and run” – making all three posts in the same visit. You need multiple visits to the Discussions in order to gain multiple perspectives by reading all of the posts and replies.

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