Discussion:
We flipped coins to learn some features and facts about probabilities. We flipped over 9,000 coins with each student flipping about 30 and the combined team had about 3,000 and three teams brought it up to 9,000. We discovered that the distribution tended to become normal as the number of flips increased. So now I wish for you to see how this works by visiting the Rice Virtual Lab to see how a biased coin may disrupt normalcy.
There is an applet that you can use to see the results of a coin toss experiment for a perfect coin (probability of H or T with p = 0.5) and what may happen if the coin is biased and p= < 0.5 or > than 0.5.
The Rice Virtual Lab in Statistics has an applet that shows the results of the "Feynman Experiment".
In this demonstration you can specify the number of events (n) and the probability of success for any one event (p) and push the "OK" button. The initial graph shows the probability distribution associated with flipping a fair coin 12 times defining a head as a success. The vertical lines represent the probabilities of obtaining each of the 13 possible outcomes (0-12 heads). As you would expect, the most likely outcome is 6 heads. This probability distribution is called the binomial distribution.
The blue line in the graph represents the normal approximation to the binomial distribution. It is a very good approximation in this case. The higher the value of n and the closer p is to .5, the better the approximation will be.
You can vary n and p and investigate their effects on the sampling distribution and the normal approximation to it. Try setting n = to 30, as we did with the Feynman Experiment. You will then see that the most likely outcome is 15 and the other 31 possible results (0-30 heads) are evenly distributed around the "mean" of 15.
Try several variations with different values of p to form a skewed distribution.
Discussion:
When a coin is flipped, the outcome is either a head or a tail; when a magician guesses the card selected from a deck, the magician can either be correct or incorrect; when a baby is born, the baby is either born in the month of March or is not. In each of these examples, an event has two mutually exclusive possible outcomes. Mutually exclusive means that it is possible for only one event to occur. In a coin toss you can expect only a Head (H) or a Tail (T) but never both. For convenience, one of the outcomes can be labeled "success" and the other outcome "failure." If an event occurs n times (for example, a coin is flipped n times), then the binomial distribution can be used to determine the probability of obtaining exactly r successes in the n outcomes. The formula for the binomial probability for obtaining r successes in n trials is given in Chapter 7 of textbook on page 276. In addition, there is discussion of using Excel to compute some of these binomial outcomes on starting on page 305.
In the formula p(x) is the probability of exactly r successes, n is the number of events, and p (shown there as the Greek letter pi) is the probability of success on any one trial. There is some unusual notation included that uses !. We need to define:
n! as = n*(n-1)*(n-2)*(n-3)... This n! is called a factorial and can be a very useful idea in statistics.
This equation assumes that the events:
(a) are dichotomous (fall into only two categories H or T),
(b) are mutually exclusive,
(c) are independent, and
(d) are randomly selected.