Probability In the November election, there will be lots of talks about polls and statistics. Every so often, polling companies determine President Bush’s approval rating. The Bureau of Labor Statistics tracks how many jobs are lost or created every month, and also the unemployment rate. What do all these numbers mean? More importantly, how are they determined?

Polling and statistics are based on the laws of probability. Flip a coin. There is a 50/50 chance that it will come up heads, and a 50/50 chance that it will come up tails. The probability that it will be heads is 1:2. Q: What is the probability of rolling a 5 with a regular die? A: 1:6 Q: What is the probability of drawing a spade from a deck of cards? A: 1:4 (There are 13 spades in a deck of 52. If you drew every card, you would have drawn 13 spades, or 1 every 4 cards)

To figure out the probability of a complex event, such as rolling two or more dice you need to know how to use exponents.

Exponents and Factorials An exponent shows the number of times a number is multiplied by itself. The number appears above and to the right of the base number and is often called the “power” of that number.

62 = 36 or 6 × 6 or 6 to the second power or 6 “squared”33 = 27 or 3 × 3 × 3 or 3 to the third power or 3 “cubed”16½ = 4 or the “square root” of 162-1 = ½ A negative exponent is the inverse or 1/x of the original number, x.90 = 1 because any number to the 0 power is 1

Q: What is probability of rolling an 8 with two dice? A: 5:36 Why? There are 36 possible outcomes of rolling two dice, or 62 (6 is the number of outcomes for one die and 2 is the number of dice). 5 of those outcomes equal 8.

A factorial is the product of all factors from 1 to n. It’s used to measure a sequence. 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Q: How many different ways can you select 4 marbles from a bag? A: 24 Why? You can’t draw the same marble more than once. So there are four options for the first marble, three for the second, two for the third, and one for the fourth. Multiplying these four numbers is the same as 4! or 24.

Statistics A statistical experiment is a procedure that produces one out of many possible outcomes. The possible outcomes are known, but each individual outcome is not. When you flip a coin, heads or tails are the possible outcome.

If you flip a coin a million times and randomly select 100 flips (this is called a sample), you are likely to select 50 heads and 50 tails, or at least very close. The more flips you select (completely randomly), the closer you’ll get to 50/50.

So when pollsters say 52 percent of Americans approve of the job President Bush is doing, they’re really saying that 52 percent of a sample of Americans approve. But because of the laws of probability, this sample is very close to the actual percentage of all Americans.

Not all polls are perfect, and you’ll notice every poll has a margin of error. If the margin of error is +/- 4%, then it really means that the approval rating is between 48 percent and 56 percent. This takes into account the fact that the sample might not be perfectly random and has some kind of sampling bias.

Graphs When organizing information, you will want to use a graph. There are several types of graphs (Can we get graphics for this?):

A bar graph shows how multiple variables differ from each other in quantity. A pie chart shows the proportion, or percent, of one variable compared with other similar variables. A line graph shows how one variable changes over time. A scatter plot and line graph shows how a number of samples correlate between two variables.

By Matthew DeFour

• What is the probability of rolling 9 with two 8-sided dice?
• A bag has six marbles: one red, three blue, two green and one yellow.

a. What is the probability of drawing a blue marble? b. How many different ways can you draw the marbles from the bag? c. What is the probability of drawing a blue and red marble?

• A pond has 30 frogs but only 5 lily pads. How many ways can you place frogs on the lily pads?

4a. How many 5-card hands are possible using a deck of 52 cards?4b. What is the probability of drawing three spades and two diamonds?4c. What is the probability of drawing a Royal Flush (A, K, Q, J, 10 of the same suit)?4d. What is the probability of drawing 4 aces?

• .125
• 17,100,720

4a. 2,598,9604b. .000334c. .0000015