### Probability Models

**probability model**is a list of all possible outcomes of an experiment with their corresponding probabilities. The model can then be used to find probabilities of different events.

Sum | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |

Frequency | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |

Sum | $8$ | $9$ | $10$ | $11$ | $12$ |

Frequency | $5$ | $4$ | $3$ | $2$ | $1$ |

Sum | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ |

Frequency | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ |

$P(\text{sum)}$ | $\frac{1}{36}$ | $\frac{2}{36}=\frac{1}{18}$ | $\frac{3}{36}=\frac{1}{12}$ | $\frac{4}{36}=\frac{1}{9}$ | $\frac{5}{36}$ | $\frac{6}{36}=\frac{1}{6}$ |

Sum | $8$ | $9$ | $10$ | $11$ | $12$ |

Frequency | $5$ | $4$ | $3$ | $2$ | $1$ |

$P(\text{sum)}$ | $\frac{5}{36}$ | $\frac{4}{36}=\frac{1}{9}$ | $\frac{3}{36}=\frac{1}{12}$ | $\frac{2}{36}=\frac{1}{18}$ | $\frac{1}{36}$ |

### Independent and Dependent Events

Two events are **independent events** if the result of one event does not affect the probabilities of the other event. Rolling a die and flipping a coin are independent events, since the result of one does not affect the result of the other. When a coin is flipped twice, the two events are independent since the probabilities for the second flip are not affected by the result of the first flip.

Two events are **dependent events** if the result of one event affects the probabilities of another event. The events of choosing a marble from a bag and choosing a second marble without replacing the first are dependent because the first marble chosen changes the number and kind of marbles remaining in the bag. The probabilities for the second marble chosen are affected by choosing the first marble.

Note that if two marbles are chosen from a bag and the first marble is replaced after being chosen, then the events of choosing two marbles from a bag are independent since the probabilities for the second marble are not affected by the result of choosing the first marble. Look for phrases like "with replacement" or "without replacement" when determining whether events are independent or dependent.

Determine if the events are independent or dependent.

A random number generator will select each number at random, and selecting the first does not affect selecting the second. So, these are independent events and this formula can be used:Identify the number of ways each event can occur and the number of equally likely outcomes.

There are 20 numbers that may be chosen in each event.

Determine if the events are independent or dependent.

When the first designer chooses a token, there are six tokens in the bag. The token that the first designer chooses will not be replaced, so when the second designer chooses a token, there will only be five tokens in the bag.

Calculate the number of ways each event can occur and the number of equally likely outcomes for each event.

Let $A$ be the event of the first designer choosing a token labeled spring collection. When the first designer chooses a style, there are three tokens labeled with spring collection in a bag with six tokens.### Mutually Exclusive Events

Two events in an experiment that cannot occur at the same time are **mutually exclusive events**. For example, when flipping a coin, the outcome is either heads or tails—it cannot be both. When flipping a coin four times, the events of flipping a head exactly three times and flipping a tail exactly four times are mutually exclusive since they cannot happen at the same time.

### Mutually Inclusive Events

Two events in an experiment that may occur at the same time are **mutually inclusive events**. For example, when rolling a die, the event of rolling a multiple of two and the event of rolling a multiple of three can occur at the same time when the outcome of the roll is six.

Consider the sample spaces for each event in rolling a multiple of two or a multiple of three. The sample space for the event of rolling a multiple of two is $\{2, 4, 6\}$. The sample space for the event of rolling a multiple of three is $\{3, 6\}$. If each outcome in the two sample spaces is counted, the outcome of rolling a 6 would be counted twice. To calculate the probability, the outcomes that occur in both lists must be removed so that they are not counted more than once.

Calculate the number of ways each event can occur and the number of equally likely outcomes for each event. Calculate the number of outcomes that occur for both events.

Let $A$ be the event of having a cat. There are 30 people out of 100 who have a cat.### Types of Probability Situations

When calculating the probability of two events, first identify how the events are related. Ask questions like:

- Does the experiment or situation involve performing two actions?
- If the results of the first action do not affect what can happen in the second action, then the events are independent: $P(A\text{ and }B)=P(A) \cdot P(B)$.
- If the results of the first action affect what can happen in the second action, then the events are dependent: $P(A\text{ and }B)=P(A) \cdot P(B \text{ after } A)$.

- Does the experiment or situation involve performing one action for which the outcome may have more than one characteristic?
- If the characteristics cannot occur at the same time, then the events are mutually exclusive: $P(A \text{ or } B) = P(A) + P(B)$.
- If the characteristics may occur at the same time, then the events are mutually inclusive: $P(A\text{ or }B)=P(A) + P(B) - P(A \text{ and } B)$.

For example, in a game that uses lettered tiles:

- The probability of drawing two tiles with vowels if the player draws a tile, puts it back, and then draws a second tile involves independent events.
- The probability of drawing two tiles with vowels if the player draws a tile, keeps it, and then draws a second tile involves dependent events.
- The probability of drawing a tile that is a blank tile or a vowel involves mutually exclusive events.
- The probability of drawing a tile that is in the word "vowel" or in the word "consonant" involves mutually inclusive events.