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What is probability?

- In your own words…
- Probability of an event is the number from 0 to 1 that indicates the likelihood the event will occur.
- An event is an outcome or collection of outcomes
- Probabilities can be written as a decimal, a percent, or a fraction.

What is probability?

- For example, when rolling a dice there are six possible outcomes or events: 1, 2, 3, 4, 5, or 6.
- A different event could be “rolling an odd number”. So, the event would be 1, 3, or 5

Probability

- Using a number line can help describe the probability of an event.

Event is more likely not to occur

Event is more likely to occur

P = 1

Event is certain

to occur

P = 0

Event will not occur

P = .5

Even is equally

likely to occur

or not occur

Theoretical Probability

- When all outcomes are equally likely, the theoretical probability that an event A will occur is:
- P(A) = number of outcomes in event A ÷ Total number of outcomes
- The theoretical probability of an even tis often simply called the probability of the event.

Find Probabilities of an Event

- You roll a standard six-sided die. Find to probability of:
- Rolling a 5
- Rolling an even number

Find Probabilities of an Event

- There are 6 possible outcomes. Only 1 outcome corresponds to rolling a 5.

P(rolling a 5) = =

# of ways to roll a 5

# of ways to roll the die

Find Probabilities of an Event

- Find the probability of rolling an even number.
- A total of 3 outcomes correspond to rolling an even number: a 2, 4, or 6

P(rolling even number) =

=

# of ways to roll an even number

# of ways to roll the die

Find Probabilities of an Event

- You have an equally likely chance of choosing any integer from 1 through 20. Find the probability of the given event.
- A perfect square is chosen.
- A factor of 30 is chosen.

Answers

- A perfect square is chosen
- A factor of 30 is chosen

Use Permutations or Combinations

- A community center hosts a talent contest for local musicians. On a given evening, 7 musicians are scheduled to perform. The order in which the musicians perform is randomly selected during the show.

Use Permutations or Combinations

- What is the probability that the musicians perform in alphabetical order by their last names? (Assume that no two musicians have the same last name)

Use Permutations or Combinations

- There are 7! different permutations of the 7 musicians. Of these, only 1 is in the alphabetical order by last name.
- So the probability of is:

P(alphabetical order) =

Use Permutation or Combinations

- You are friends with 4 of the musicians. What is the probability that the first 2 performers are your friends?

Use Permutations and Combinations

- There are 7C2 different combinations of 2 musicians. Of these, 4C2 are 2 of your friends.
- So, the probability is:

P(first 2 performers are your friends) =

Odds

- You can also use odds to measure the likelihood that an event will occur.
- Odds measure the chances in favor of an event occurring or the chances against an event occurring.

Odds

- When all outcomes are equally likely, the odds in favor and not in favor of an event A are as follows:

Odds in favor of event A =

Odds not in favor of event A =

- You can write the odds in favor of or against an event in the form or in the form a:b.

# of outcomes in A

# of outcomes not in A

# of outcomes not in A

# of outcomes in A

Find odds

- A card is drawn from a standard deck of 52 cards. Find:
- The odds in favor of drawing a 10
- The odds against drawing a club

Find odds

- Odds in favor of drawing a 10 =

= or 1:12

- Odds against drawing a club =

= or 3:1

# of non-clubs

# of tens

# of non-tens

# of clubs

Find Odds

- What are the odds:
- In favor of drawing a heart
- Against drawing a queen

Experimental Probability

- Sometimes it is not possible or convenient to find the theoretical probability of an event.
- In such cases, you may be able to calculate an experimental probability by performing an experiment, conducting a survey, or looking at the history of the event.

Experimental Probability

- When an experiment is performed that consists of a certain number of trials, the experimental probability of an event A is given by:

P(A) =

# of trials where A occurs

Total # of trials

Geometric Probability

- Some probabilities are found by calculating a ratio of two lengths, areas, or volumes.
- These are called geometric probabilities

Geometric Probability

- Length
- Area
- Volume

area of smaller object

volume of smaller space

length of smaller segment

total length

total area

total volume

Geometric Probability

- Darts and Bulls-eye

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