Binomial Chart

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Binomial Distribution

BINOMIAL DISTRIBUTION

The mathlet above displays the probability distribution of any number of independent binary events. Input the number of trials, the chance for each success, the height of the tallest bar graph, and the maximum and minimum for the range.

Initially, the mathlet displays the probabilities of flipping x number of heads in 10 tosses; and, also, the probabilities of x number of boys in a family with 10 children. Remember, if a distribution is binomial, then there are only two outcomes: heads or tails, boy or girl, yes or no, true or false, right or wrong, win or lose. Each probability is the complement of the other. Also each trial run is independent of all the other trials. Many of our coin, card, and dice experiments were examples of binomial probability.

For our ESP game, input 5 for the number n of trials (number of cards). Input .25 for the chance p of guessing a suit. Input 1200 for the height of the tallest bar. Input 5 for the maximum. The numbers on the bottom ( 0 to 5) represent the number of possible correct guesses. The number on top of each bar is the probability of correctly guessing the number on the bottom.

For our 4 dice game, input 4 for the number n of trials (number of dice). Input .1667 for the chance p=1/6 of rolling a six on one die. Input 900 for the height of the tallest bar. Input 4 for the maximum. The numbers on the bottom ( 0 to 4) are the probabilities of rolling 0, 1, 2, 3, and 4 sixes, respectively. The number on top of each bar is the probability of correctly guessing the number on the bottom.

When measuring the probability distribution of binary events, the Variance can be approximated as the product of the number of trials, the probability of a win, and the probability of a loss: VAR = npq. The square root of the Variance is the Standard Deviation. Likewise, the Mean, or Average, can be approximated as the product of the number of trials and the probability of a win: AVG = np.

FOR DISCUSSION

To toss 100 coins:
1. Input 100 for Trials
2. Input .5 for P(win)
3. Input 6500 for Height
4. Input 100 for Maximum
5. Press the ENTER key
6. What do you see?
7. What do you not see?
In the upper left corner of the mathlet are the expected Average and Standard Deviation for our coin toss.
How many Heads should we expect to toss on average in 100 flips?
What is the expected Standard Deviation?
What is the meaning of the Standard Deviation?
How many "stair steps" can you count in the graph?
The Standard Deviation tells us how many of those steps equal 1 Standard Deviation.
How many standard deviations are visible on the graph?
Answer: divide the number of red steps by the Standard Deviation!
What is the range of visible red steps?
What number of coins is represented by the shortest red step on the left?
Enter this number in Minimum box.
What number of coins is represented by the shortest red step on the right?
Enter this number in Maximum box. Press ENTER key and zoom in on the curve.
Subtract the Standard Deviation from the Average.
This number is -1 Standard Deviations from the Average.
Add the Standard Deviation from the Average.
This number is +1 Standard Deviations from the Average.
What is sum of the probabilities between -1 and +1 Standard Deviations?
What is sum of the probabilities between -2 and +2 Standard Deviations?
What is sum of the probabilities between -3 and +3 Standard Deviations?
Compare these sums to the probabilities in the Bell Curve at the top of the page?
What are the differences between them?
Why do you think there are differences between our bar chart and the Bell Curve?
Increase the number of coins to 1000.
1. Enter 1000 in the Trials box
2. Enter 2000 in the Height box
3. Enter 0 in the Minimum box
4. Enter 1000 in the Maximum box
5. Press the ENTER key.
What happens to the shape of our coin-tossing bar chart as the number of coins increases?
How many coins would we need to toss before the probabilities in our bar chart equal those in the Bell Curve?

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Last Modified 11/13/08 9:19 AM