13.7 Probability

Tossing a coin:

1. ⓐ Describe in set notation the sample space of tossing a coin.
2. ⓑ Find the probability of “Coin lands on heads.”

Rolling a die:

1. ⓐ Describe in set notation the event “Rolling an odd number.”
2. ⓑ Find the probability of “Rolling an odd number.”

Drawing a card:

1. ⓐ Describe in set notation the event “Drawing an Ace.”
2. ⓑ Find the probability of “Drawing an Ace.”

Construct a probability model for rolling a single, fair die, with the event being the number shown on the die.

Construct a probability model for tossing a fair coin.

A six-sided number cube is rolled. Find the probability of rolling an odd number.

A number cube is rolled. Find the probability of rolling a number greater than 2.

A card is drawn from a standard deck. Find the probability of drawing a heart or a 7.

A card is drawn from a standard deck. Find the probability of drawing a red card or an ace.

A card is drawn from a standard deck. Find the probability of drawing a heart or a spade.

A card is drawn from a standard deck. Find the probability of drawing an ace or a king.

Two six-sided number cubes are rolled.

1. ⓐFind the probability that the sum of the numbers rolled is less than or equal to 3.
2. ⓑFind the probability that the sum of the numbers rolled is greater than 3.

Two number cubes are rolled. Use the Complement Rule to find the probability that the sum is less than 10.

A child randomly selects 5 toys from a bin containing 3 bunnies, 5 dogs, and 6 bears.

1. ⓐFind the probability that only bears are chosen.
2. ⓑFind the probability that 2 bears and 3 dogs are chosen.
3. ⓒFind the probability that at least 2 dogs are chosen.

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.

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.

What is an experiment?

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?

Landing on a vowel

Landing on purple or a vowel

Landing on green or blue

Not landing on yellow or a consonant

Two coins are tossed.

Find the probability of tossing two heads.

Two coins are tossed.

Find the probability of tossing at least one tail.

Four coins are tossed.

Find the probability of tossing exactly two heads.

Four coins are tossed.

Find the probability of tossing four heads or four tails.

Four coins are tossed.

Find the probability of tossing not all tails.

Four coins are tossed.

Find the probability of tossing either two heads or three heads.

One card is drawn from a standard deck of $52$ cards. Find the probability of drawing the following:

A two

One card is drawn from a standard deck of $52$ cards. Find the probability of drawing the following:

Red six

One card is drawn from a standard deck of $52$ cards. Find the probability of drawing the following:

A non-ace

Two dice are rolled, and the results are summed.

Construct a table showing the sample space of outcomes and sums.

Two dice are rolled, and the results are summed.

Find the probability of rolling at least one four or a sum of $8.$

Two dice are rolled, and the results are summed.

Find the probability of rolling a sum greater than or equal to $15.$

Two dice are rolled, and the results are summed.

Find the probability of rolling a sum less than $6$ or greater than $9.$

Two dice are rolled, and the results are summed.

Find the probability of rolling a sum of $5$ or $6.$

A coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:

A head on the coin or a club

A coin is tossed, and a card is pulled from a standard deck. Find the probability of the following:

A head on the coin or a face card

Use this scenario: a bag of M&Ms contains $12$ blue, $6$ brown, $10$ orange, $8$ yellow, $8$ red, and $4$ green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.

What is the probability of getting all blue M&Ms?

Use this scenario: a bag of M&Ms contains $12$ blue, $6$ brown, $10$ orange, $8$ yellow, $8$ red, and $4$ green M&Ms. Reaching into the bag, a person grabs 5 M&Ms.

What is the probability of getting $3$ blue M&Ms?

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting $20$ numbers from the numbers $1$ to $80.$ After the player makes his selections, $20$ winning numbers are randomly selected from numbers $1$ to $80.$ A win occurs if the player has correctly selected $3 , 4 ,$ or $5$ of the $20$ winning numbers. (Round all answers to the nearest hundredth of a percent.)

What is the percen…

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting $20$ numbers from the numbers $1$ to $80.$ After the player makes his selections, $20$ winning numbers are randomly selected from numbers $1$ to $80.$ A win occurs if the player has correctly selected $3 , 4 ,$ or $5$ of the $20$ winning numbers. (Round all answers to the nearest hundredth of a percent.)

What is the percen…

Use the following scenario for the exercises that follow: In the game of Keno, a player starts by selecting $20$ numbers from the numbers $1$ to $80.$ After the player makes his selections, $20$ winning numbers are randomly selected from numbers $1$ to $80.$ A win occurs if the player has correctly selected $3 , 4 ,$ or $5$ of the $20$ winning numbers. (Round all answers to the nearest hundredth of a percent.)

How much less is…

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).United States Census Bureau. http://www.census.gov

If you meet five U.S. citizens, what is the percent chance that exactly one is elderly? (Round to the nearest tenth of a percent.)

Use this data for the exercises that follow: In 2013, there were roughly 317 million citizens in the United States, and about 40 million were elderly (aged 65 and over).United States Census Bureau. http://www.census.gov

If you meet five U.S. citizens, what is the percent chance that four are elderly? (Round to the nearest thousandth of a percent.)

Write the first four terms of the sequence defined by the recursive formula $a_{1} = 2 , a_{n} = a_{n - 1} + n .$

Write the first four terms of the sequence defined by the explicit formula $a_{n} = 10^{n} + 3.$

Is the sequence $\frac{4}{7} , \frac{47}{21} , \frac{82}{21} , \frac{39}{7} , ...$ arithmetic? If so, find the common difference.

An arithmetic sequence has the first term $a_{1} = 18$ and common difference $d = - 8.$ What are the first five terms?

Write a recursive formula for the arithmetic sequence $- 20 , - 10 , 0 , 10 ,\ldots$

Write an explicit formula for the arithmetic sequence $\frac{7}{8} , \frac{29}{24} , \frac{37}{24} , \frac{15}{8} , \ldots$

Find the common ratio for the geometric sequence $2.5 , 5 , 10 , 20 , \ldots$

A geometric sequence has terms $a_{7} = 16 , 384$ and $a_{9} = 262 , 144$. What are the first five terms?

What are the first five terms of the geometric sequence $a_{1} = 3 , a_{n} = 4 \cdot a_{n - 1} ?$

Write an explicit formula for the geometric sequence $- \frac{1}{5} , - \frac{1}{15} , - \frac{1}{45} , - \frac{1}{135} , \ldots$

Use summation notation to write the sum of terms $\frac{1}{2} m + 5$ from $m = 0$ to $m = 5.$

Use the formula for the sum of the first $n$ terms of an arithmetic series to find the sum of the first eleven terms of the arithmetic series 2.5, 4, 5.5, … .

Use the formula for the sum of the first n terms of a geometric series to find $S_{9}$ for the series $12 , 6 , 3 , \frac{3}{2} , \ldots$

Find the sum of the infinite geometric series $\sum_{k = 1}^{\infty} 45 \cdot \left(- \frac{1}{3}\right)^{k - 1} .$

Alejandro deposits $80 of his monthly earnings into an annuity that earns 6.25% annual interest, compounded monthly. How much money will he have saved after 5 years?

How many ways are there to choose a number from the set $\left\{- 10 , - 6 ,\text{ } 4 ,\text{ } 10 ,\text{ } 12 ,\text{ } 18 ,\text{ } 24 ,\text{ } 32\right\}$ that is divisible by either $4$ or $6 ?$

How many ways are there to construct a 4-digit code if numbers can be repeated?

Calculate $P \left(18 , 4\right) .$

Calculate $C \left(15 , 6\right) .$

How many subsets does the set $\left\{1 , 3 , 5 , \ldots , 99\right\}$ have?

How many distinct ways can the word DEADWOOD be arranged?

Evaluate the binomial coefficient $\begin{pmatrix} 23 \\ 8 \end{pmatrix} .$

Use the Binomial Theorem to write the first three terms of $\left(2 a + b\right)^{17} .$

Assume two die are rolled.

Construct a table showing the sample space.

Assume two die are rolled.

What is the probability of rolling a pair?

Assume two die are rolled.

What is the probability that a roll doesn’t include a 2 or result in a pair?

Assume two die are rolled.

What is the probability that a roll includes neither a 5 nor a 6?

Use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

What is the percent chance that at least one of the children attending the party prefers milk?

Use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

What is the percent chance that exactly 3 of the children attending the party prefer milk?

Write the first four terms of the sequence defined by the recursive formula $a = – 14 , a_{n} = \frac{2 + a_{n – 1}}{2} .$

Is the sequence $0.3 , 1.2 , 2.1 , 3 , \ldots$ arithmetic? If so find the common difference.

Write a recursive formula for the arithmetic sequence $- 2 , - \frac{7}{2} , - 5 , - \frac{13}{2} , \ldots$ and then find the 22^nd term.

Is the sequence $- 2 , - 1 , - \frac{1}{2} , - \frac{1}{4} , \ldots$ geometric? If so find the common ratio. If not, explain why.

Write a recursive formula for the geometric sequence $1 , - \frac{1}{2} , \frac{1}{4} , - \frac{1}{8} , \ldots$

Use summation notation to write the sum of terms $3 k^{2} - \frac{5}{6} k$ from $k = - 3$ to $k = 15.$

Use the formula for the sum of the first $n$ terms of a geometric series to find $\sum_{k = 1}^{7} - 0.2 \cdot \left(- 5\right)^{k - 1} .$

Ramla deposits $3,600 into a retirement fund each year. The fund earns 7.5% annual interest, compounded monthly. If she opened her account when she was 20 years old, how much will she have by the time she’s 55? How much of that amount was interest earned?

A buyer of a new sedan can custom order the car by choosing from 5 different exterior colors, 3 different interior colors, 2 sound systems, 3 motor designs, and either manual or automatic transmission. How many choices does the buyer have?

A music group needs to choose 3 songs to play at the annual Battle of the Bands. How many ways can they choose their set if they have 15 songs to pick from?

How many distinct ways can the word EVANESCENCE be arranged if the anagram must end with the letter E?

Find the seventh term of $\left(x^{2} - \frac{1}{2}\right)^{13}$ without fully expanding the binomial.

What is the probability of landing on an odd number?

What is the probability of landing on blue or an odd number?

A bowl of candy holds 16 peppermint, 14 butterscotch, and 10 strawberry flavored candies. Suppose a person grabs a handful of 7 candies. What is the percent chance that exactly 3 are butterscotch? (Show calculations and round to the nearest tenth of a percent.)