Toss 3 coins. Find the probability that at least one head shows by writing out the sample space

Sample space = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}; probability of at least one head is then $7/8=0.875$Is it unusual to get 3 heads when 3 coins are tossed?

The probability of 3 heads is $1/8 = 0.125$, which is not terribly unusual. (*Typically, events that have probability less than $0.05$ are considered unusual.*)In rolling a single die, let $A$ represent rolling an even value and $B$ represent rolling a value greater than 4. Find $P(A \textrm{ or } B)$ in two ways: a) using the addition rule and b) considering the sample space.

There are 4 out of 6 values in the sample space that fit this criteria (i.e., $\{2,4,5,6\}$), so the probability is $4/6 = 2/3$.

Using the addition rule, we have: $P(\textrm{even}) + P(\gt4) - P(\textrm{even and } \gt 4) = 3/6 + 2/6 - 1/6 = 4/6 = 2/3$

In rolling a single die, let $A$ represent rolling an odd value and $B$ represent rolling a $6$. Find $P(A \textrm{ and } B)$ in two ways: a) using the multiplication rule and b) considering the sample space.

A and B can't both happen. A number can't be both odd and $6$. So the probability is $0$.A single card is drawn from a deck. Find the probability of selecting:

- a 4 or a diamond
- a club or a diamond
- a jack or a black card

$\displaystyle{\frac{4}{52} + \frac{13}{52} - \frac{1}{52} \doteq 0.3077}$

$\displaystyle{\frac{13}{52} + \frac{13}{52} = 0.5}$

$\displaystyle{\frac{4}{52} + \frac{26}{52} - \frac{2}{52} \doteq 0.5385}$

At a used-book sale, 100 books are adult books and 160 are children's books. Of the adult books, 70 are nonfiction while 60 of the children's books are nonfiction. If a book is selected at random, find the probability that it is:

- fiction
- not a children's nonfiction book
- an adult book or a children's nonfiction book

$\displaystyle{\frac{130}{260} = 0.5}$

$\displaystyle{1 - \frac{60}{260} \doteq 0.2308}$

$\displaystyle{\frac{100}{260} + \frac{60}{260} \doteq 0.6154}$

When two dice are rolled, find the probability of getting a sum that is

- 5 or 6
- greater than 9
- less than 4 or greater than 9
- divisible by 4

$\displaystyle{\frac{4+5}{36} = \frac{1}{4}}$

$\displaystyle{\frac{3+2+1}{36} = \frac{1}{6}}$

$\displaystyle{\frac{1+2+3+2+1}{36} = \frac{1}{4}}$

$\displaystyle{\frac{3+5+1}{36} = \frac{1}{4}}$

Suppose two coins are tossed, and you know that at least one of them resulted in "tails". What is the probability that they are both tails?

$\displaystyle{\frac{1}{3}}$In drawing two cards from a standard deck, what is the probability of drawing an ace on the first draw and a king on the second draw?

$\displaystyle{\frac{4}{52} \cdot \frac{4}{51} \doteq 0.0060}$Roll a single die. What is the probability that one rolls a 1 or 2 given that one rolled an even value?

$\displaystyle{\frac{1}{3}}$At a large university, the probability that a student takes calculus and is on the dean's list is 0.042. The probability that a student is on the dean's list is 0.21. Find the probability that the student is taking calculus given that he or she is on the dean's list.

$\displaystyle{\frac{0.042}{0.21} = 0.2}$Eighty students in a school cafeteria were asked if they favored a ban on smoking in the cafeteria. The results of the survey are shown in the table below: $$\begin{array}{l|c|c|c} & \textrm{Favor} & \textrm{Oppose} & \textrm{No Opinion}\\\hline \textrm{Freshmen} & 15 & 27 & 8\\\hline \textrm{Sophomore} & 23 & 5 & 2\\\hline \end{array}$$ If a student is selected at random, find the probability that

- given the student is a freshman, he or she opposes the ban
- given the student favors the ban, he or she is a sophomore

$\displaystyle{\frac{27}{15+27+8} = 0.54}$

$\displaystyle{\frac{23}{15+23} \doteq 0.6053}$

Find the probability that if a coin is tossed twice, the first toss is "heads", while the second is "tails"

$\displaystyle{\frac{1}{2} \cdot \frac{1}{2} = 0.25}$If 37% of high school students said that they exercise regularly, find the probability that 5 randomly selected high school students will say that they exercise regularly.

$\displaystyle{(0.37)^5 \doteq 0.0069}$If 2 cards are selected from a standard deck of 52 cards without replacement, find the probability that

- both are spades
- both are the same suit

$\displaystyle{\frac{13}{52} \cdot \frac{12}{51} \doteq 0.0588}$

$\displaystyle{\frac{12}{52} \doteq 0.2308}$

The U.S. Department of Health and Human Services reports that 15% of Americans have chronic sinusitis. If 5 people are selected at random, find the probability that at least one has chronic sinusitis.

Use the complement. $\quad 1 - (0.85)(0.85)(0.85)(0.85)(0.85) = 1 - 0.85^5 \doteq 0.5563$A jar contains 8 red marbles, 9 blue marbles, and 10 green marbles. Four marbles are chosen at random without replacement. Find the probability of getting

- all green marbles
- 2 red and 2 blue marbles
- no green marbles
- exactly 2 green marbles
- at most 2 green marbles
- marbles that are all the same color

$\displaystyle{\frac{10}{27} \cdot \frac{9}{26} \cdot \frac{8}{25} \quad \textrm{or} \quad \frac{({}_{10} C_4)}{({}_{27} C_4)}}$

$\displaystyle{\frac{({}_8 C_2)({}_9 C_2)}{({}_{27} C_4)}}$

$\displaystyle{\frac{17}{27} \cdot \frac{16}{27} \cdot \frac{15}{27} \quad \textrm{or} \quad \frac{({}_{17} C_4)}{({}_{27} C_4)}}$

$\displaystyle{\frac{({}_{10} C_2)({}_{17} C_2)}{({}_{27} C_4)}}$

$\displaystyle{\frac{({}_{10} C_0)({}_{17} C_4)}{({}_{27} C_4)} + \frac{({}_{10} C_1)({}_{17} C_3)}{({}_{27} C_4)} + \frac{({}_{10} C_2)({}_{17} C_2)}{({}_{27} C_4)}}$

$\displaystyle{\frac{({}_{8} C_4)}{({}_{27} C_4)} + \frac{({}_{9} C_4)}{({}_{27} C_4)} + \frac{({}_{10} C_4)}{({}_{27} C_4)}}$

An automobile manufacturer has three factories: A, B, and C. They produce 50%, 30%, and 20% respectively, of a specific model of car. 30% of the cars produced in factory A are white, 40% of those produced in factory B are white, and 25% produced in factory C are white.

- If an automobile produced by the company is selected at random, find the probability that it is white.
- Given that an automobile selected at random is white, find the probability that it came from factory B.

Two manufacturers supply blankets to emergency relief organizations. Manufacturer A supplies 3000 blankets and 4% are irregular in workmanship. Manufacturer B supplies 2400 blankets and 7% are found to be irregular. Given that a blanket is irregular, find the probability that it came from manufacturer B.

**The Birthday Problem.**Find the probability that 2 or more people in a group of $n$ people have the same birthday.In the game of Craps, a player rolls a pair of dice. If, on this first roll, the sum of the dice is 7 or 11, they win immediately. If the sum of the dice is 2, 3, or 12, they lose immediately. If instead, they roll some other value $x$ (called the "point"), they continue to roll the dice until either the point is rolled (where the player then wins), or a 7 is rolled (where the player then loses). What is the probability that the player wins a game of Craps?

In rolling a pair of dice, what is the probability that you roll

- at least a sum of 7?
- at most a 7 given that at least one die shows a 3

Toss a coin 3 times. What is the probability that

- there is at most one head
- one gets 2 heads given that the first toss is a tail

For a bridge hand of 13 cards, find the probability that you have

- exactly 9 spades
- all 4 aces and no face cards
- 5 spades, 6 hearts, and 2 diamonds, given that none of your cards are aces

A 6-sided die is rolled 4 times. Find the probability of rolling at least one 6.

Two 6-sided dice are rolled 24 times. Find the probability of rolling at least one sum of 12 (i.e., two sixes).