There are two properties that define Pascal's Triangle:
•
Any number on the edge of Pascal's Triangle is 1.
•
Any other number is found by adding the two numbers above it.
For example, if you add the 1 and the 3 that start the third row, you get 4, which you place below these two numbers. You've now started a fourth row. Then you add 3 and 3 to get 6, and 3 and 1 to get 4, which are the next two values in the fourth row, and so forth.
Video Segment In this video segment, Professor Kader introduces Pascal's Triangle. Watch this segment to review the process of generating Pascal's Triangle. As you watch, keep in mind the following questions:
•
What does each number in Pascal's Triangle represent?
•
Why is Pascal's Triangle useful?
If you're using a VCR, you can find this segment on the session video approximately 16 minutes and 31 seconds after the Annenberg Media logo.
In this fashion, we can use successive rows of Pascal's Triangle to predict the frequencies for the number of heads in binomial experiments with increasing numbers of trials.
Note: The first number in a row of Pascal's Triangle corresponds to the frequency of zero heads in a number of coin tosses, not one; therefore, the third number in a row corresponds to the frequency of two, and not three, heads. This is a very common error.
Problem C6
Use the properties of Pascal's Triangle to generate the fifth and sixth rows.
Problem C7
Use Pascal's Triangle to determine the frequencies and probabilities for five and six tosses of a fair coin. If you wish, confirm your results with a tree diagram.
Write a formula for determining the number of possible outcomes of n tosses of a fair coin.
Think about the tree diagram and start with n = 1 coin toss. Then look at n = 2 tosses. Then look at n = 3 tosses. Note that as you increase the number of tosses by one, you double the number of outcomes. Close Tip
Video Segment In this video segment, Professor Kader discusses the usefulness of Pascal's Triangle for determining probabilities. He then asks participants to use Pascal's Triangle to determine the probabilities for a binomial experiment when n = 5. Watch this segment after completing Problems C7 and C8.
Note that the binomial experiment conducted by the onscreen participants involved predicting the outcomes when rolling a pair of dice.
If you're using a VCR, you can find this segment on the session video approximately 18 minutes and 30 seconds after the Annenberg Media logo.
Extend Pascal's Triangle to the 10th row. Using the 10th row, determine the probability of tossing exactly five heads out of 10 coin tosses.
There will be 1,024 total outcomes in the 10th row, so the probability will be the frequency (found in Pascal's Triangle) divided by 1,024. Close Tip
We've been investigating the binomial probability model. In a random experiment with two possible outcomes, this model can be used to describe the probability of either result. Consider, for example, a True-False test. If a test has four True-False questions, and you make an independent guess on each question, how many will you get correct? (Of course, the only thing you can say for sure is that you will get either zero, one, two, three, or four questions correct!)
Problem C10
Use the binomial probability model to determine the following:
a.
What is the most probable score you'll get?
b.
What are the least probable scores you'll get?
c.
What is the probability of getting at least two answers correct?
d.
What is the probability of getting at least three answers correct?
Remember that "at least" includes the score itself, so the probability of getting "at least two" answers correct includes two, three, four, and five. Close Tip
