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# The Basics of Probability Study Guide (page 2)

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Updated on Oct 5, 2011

## Probability with Several Outcomes

Consider an example involving several different outcomes.

Example: If a pair of dice is tossed, what is the probability of throwing a sum of 3?

Solution:

1. Make a table showing all the possible outcomes (sums) of tossing the dice:

2. Determine the number of favorable outcomes by counting the number of times the sum of 3 appears in the table: 2 times.
3. Determine the total number of possible outcomes by counting the number of entries in the table: 36.
4. Substitute 2 favorable outcomes and 36 total possible outcomes into the probability formula, and then reduce:

Throwing a 3 doesn't appear to be very likely with its probability of . Is any sum less likely than 3?

## Probabilities That Add Up to 1

Think again about the example of 2 red buttons and 3 blue buttons from the previous section, Finding Probability. The probability of picking a red button is and the probability of picking a blue button is . The sum of these probabilities is 1.

The sum of the probabilities of every possible outcome of an event is 1.

Notice that picking a blue button is equivalent to NOT picking a red button:

Thus, the probability of picking a red button plus the probability of NOT picking a red button is1:

P(Event will occur) + P(Event will NOT occur) = 1

#### Tip

When there are only two outcomes possible, A or B, the probability that outcome A will occur is the same as 1 – P(Event B).

Example:  A bag contains green chips, purple chips, and yellow chips. The probability of picking a green chip is and the probability of picking a purple chip is .What is the probability of picking a yellow chip? If there are 36 chips in the bag, how many are yellow?

Solution:

 1. The sum of all the probabilities is 1: P(green) + P(purple) + P(yellow) = 1 2. Substitute the known probabilities: 3. Solve for yellow: The probability of picking a yellow chip is . 4. Thus, of the 36 chips are yellow:

Thus, there are 15 yellow chips.

#### Tip

Gather the following coins together and put them into a box: 5 pennies, 3 nickels, 2 dimes, and 1 quarter. Without looking into the box, reach in to pull out an item. Before you touch any of the objects, figure out the probability of pulling out each item on your first reach.

## The Basics of Probability Sample Questions

1. What is the probability of throwing a sum of at least 7
2. What is the probability of throwing a sum of 7 or 11?

### Solutions to Sample Questions

#### Question 1

1. Determine the number of favorable outcomes by counting the number of table entries containing a sum of at least 7:

Sum # Entries
7 6
8 5
9 4
10 3
11 2
12

2. Determine the number of total possible outcomes by counting the number of entries in the table: 36.
3. Substitute 21 favorable outcomes and 36 total possible outcomes into the probability formula:

Since the probability exceeds , it's more likely to throw a sum of at least 7 than it is to throw a lower sum.

#### Question 2

There are two ways to solve this problem.

Solution #1:

1. Determine the number of favorable outcomes by counting the number of entries that are either 7 or 11:

Sum # Entries
7 6
11

2. You already know that the number of total possible outcomes is 36. Substituting 8 favorable outcomes and 36 total possible outcomes into the probability formula yields a probability of for throwing a 7 or 11:

Solution #2:

1. Determine two separate probabilities—P(7) and P(11)—and add them together:

Since P(7 or 11) = P(7) + P(11), we draw the following conclusion about events that don't depend on each other:

P(Event A or Event B) = P(Event A) + P(Event B)

Find practice problems and solutions for these concepts at The Basics of Probability Practice Questions.

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