# 11.7 Probability  (Page 5/18)

 Page 5 / 18

A child randomly selects 3 gumballs from a container holding 4 purple gumballs, 8 yellow gumballs, and 2 green gumballs.

1. Find the probability that all 3 gumballs selected are purple.
2. Find the probability that no yellow gumballs are selected.
3. Find the probability that at least 1 yellow gumball is selected.

Access these online resources for additional instruction and practice with probability.

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## Key equations

 probability of an event with equally likely outcomes $P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}$ probability of the union of two events $P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)-P\left(E\cap F\right)$ probability of the union of mutually exclusive events $P\left(E\cup F\right)=P\left(E\right)+P\left(F\right)$ probability of the complement of an event $P\left(E\text{'}\right)=1-P\left(E\right)$

## Key concepts

• Probability is always a number between 0 and 1, where 0 means an event is impossible and 1 means an event is certain.
• The probabilities in a probability model must sum to 1. See [link] .
• When the outcomes of an experiment are all equally likely, we can find the probability of an event by dividing the number of outcomes in the event by the total number of outcomes in the sample space for the experiment. See [link] .
• To find the probability of the union of two events, we add the probabilities of the two events and subtract the probability that both events occur simultaneously. See [link] .
• To find the probability of the union of two mutually exclusive events, we add the probabilities of each of the events. See [link] .
• The probability of the complement of an event is the difference between 1 and the probability that the event occurs. See [link] .
• In some probability problems, we need to use permutations and combinations to find the number of elements in events and sample spaces. See [link] .

## Verbal

What term is used to express the likelihood of an event occurring? Are there restrictions on its values? If so, what are they? If not, explain.

probability; The probability of an event is restricted to values between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}$ inclusive of $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}1.\text{\hspace{0.17em}}$

What is a sample space?

What is an experiment?

An experiment is an activity with an observable result.

What is the difference between events and outcomes? Give an example of both using the sample space of tossing a coin 50 times.

The union of two sets is defined as a set of elements that are present in at least one of the sets. How is this similar to the definition used for the union of two events from a probability model? How is it different?

The probability of the union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets and a union of events the union includes either or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}1.\text{\hspace{0.17em}}$

## Numeric

For the following exercises, use the spinner shown in [link] to find the probabilities indicated.

Landing on red

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
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It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
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hello
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