7.4: Practice Problems
- Page ID
- 139289
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
1. A six-sided die is rolled twice. What is the probability of showing a 2 on the first roll and an odd number on the second roll?
2. A six-sided die is rolled three times. What is the probability of showing an even number on all three rolls?
3. A six-sided die is rolled twice. What is the probability of showing a 6 on both rolls?
4. A fair coin is flipped twice. What is the probability of showing heads on both flips?
5. A die is rolled twice. What is the probability of showing a 5 on the first roll and an even number on the second roll?
6. A couple has three children. What is the probability that all three are girls?
7. Suppose that \(21 \%\) of people own dogs. If you pick two people at random, what is the probability that they both own a dog?
8. In your drawer you have 5 pairs of socks, 4 of which are white, and 11 tee shirts, 2 of which are white. If you randomly reach in and pull out a pair of socks and a tee shirt, what is the probability that both are white?
9. Bert and Ernie each have a well-shuffled standard deck of 52 cards. They each draw one card from their own deck. Compute the probability that:
a. Bert and Emie both draw an Ace.
b. Bert draws an Ace but Ernie does not.
c. Neither Bert nor Ernie draws an Ace.
d. Bert and Ernie both draw a heart.
e. Bert gets a card that is not a Jack and Emie draws a card that is not a heart.
10. Bert has a well-shuffled standard deck of 52 cards, from which he draws one card; Ernie has a 12-sided die, which he rolls at the same time Bert draws a card. Compute the probability that:
a. Bert gets a Jack and Ernie rolls a five.
b. Bert gets a heart and Ernie rolls a number less than six.
c. Bert gets a face card (Jack, Queen or King) and Ernie rolls an even number.
d. Bert gets a red card and Emie rolls a fifteen.
e. Bert gets a card that is not a Jack and Ernie rolls a number that is not twelve.
11. A jar contains 5 red marbles numbered 1 to 5 and 8 blue marbles numbered 1 to 8 . A marble is drawn at random from the jar. Find the probability the marble is
a. Even-numbered given that the marble is red.
b. Red given that the marble is even-numbered.
12. A jar contains 4 red marbles numbered 1 to 4 and 8 blue marbles numbered 1 to 8 . A marble is drawn at random from the jar. Find the probability the marble is
a. Odd-numbered given that the marble is blue.
b. Blue given that the marble is odd-numbered.
13. Compute the probability of flipping a coin and getting heads, given that the previous flip was tails.
14. Find the probability of rolling a " 1 " on a fair die, given that the last 3 rolls were all ones.
15. Suppose a math class contains 25 students, 14 females (three of whom speak French) and 11 males (two of whom speak French).
a. Compute the probability that a randomly selected student speaks French, given that the student is female.
b. Compute the probability that a randomly selected student is male, given that the student speaks French.
16. A test was given to a group of students. The grades and gender are summarized below.
|
A |
B |
C |
Total |
|
|
Male |
10 |
12 |
2 |
24 |
|
Female |
16 |
6 |
9 |
31 |
|
Total |
26 |
18 |
11 |
55 |
Suppose a student is chosen at random:
a. Find the probability that the student was male given they earned an A.
b. Find the probability that the student was male given they earned a \(\mathrm{C}\).
c. Find the probability that the student was female given they earned a B.
17. A test was given to a group of students. The grades and gender are summarized below.
|
A |
B |
C |
Total |
|
|
Male |
7 |
6 |
4 |
17 |
|
Female |
9 |
2 |
10 |
21 |
|
Total |
16 |
8 |
14 |
38 |
Suppose a student is chosen at random:
a. Find the probability that the student earned a B given they are male.
b. Find the probability that the student earned a B given they are female.
c. Find the probability that the student earned a \(\mathrm{C}\) given they are male.
18. A certain virus infects one in every 400 people. A test used to detect the virus in a person is positive \(90 \%\) of the time if the person has the virus and \(10 \%\) of the time if the person does not have the virus. Let A be the event "the person is infected" and \(\mathrm{B}\) be the event "the person tests positive".
a. Find the probability that a person has the virus given that they have tested positive, i.e. find \(\mathrm{P}(\mathrm{A} \mid \mathrm{B})\).
b. Find the probability that a person does not have the virus given that they test negative, i.e. find \(\mathrm{P}(\operatorname{not} \mathrm{A} \mid \operatorname{not} \mathrm{B})\).
19. A certain virus infects one in every 2000 people. A test used to detect the virus in a person is positive \(96 \%\) of the time if the person has the virus and \(4 \%\) of the time if the person does not have the virus. Let A be the event "the person is infected" and \(\mathrm{B}\) be the event "the person tests positive".
a. Find the probability that a person has the virus given that they have tested positive, i.e. find \(\mathrm{P}(\mathrm{A} \mid \mathrm{B})\).
b. Find the probability that a person does not have the virus given that they test negative, i.e. find \(\mathrm{P}(\operatorname{not} \mathrm{A} \mid\) not \(\mathrm{B})\).
20. Two cards are drawn from a standard deck of cards. What is the probability of drawing a King and then drawing a Queen.
21. Two cards are drawn from a standard deck of cards. What is the probability of both cards being red?
22. Two cards are drawn from a standard deck of cards. What is the probability of drawing a Jack and then drawing an Ace?
23. Five cards are drawn from a standard deck of cards. What is the probability of all cards being black?
24. Tony buys a bag of cookies that contains 4 chocolate chip cookies, 9 peanut butter cookies, 7 sugar cookies and 8 oatmeal cookies. What is the probability that Tony reaches in the bag and randomly selects a peanut butter cookie from the bag, eats it, then reaches back in the bag and randomly selects a sugar cookie?
25. Suppose a jar contains 17 red marbles and 32 blue marbles. If you reach in the jar and pull out 2 marbles at random, find the probability that both are red.
26. Suppose you write each letter of the alphabet on a different slip of paper and put the slips into a hat. If you pull out two slips at random, find the probability that both are vowels.
27. A math class consists of 25 students, 14 female and 11 male. Two students are selected at random to participate in a probability experiment. Compute the probability that
a. a male is selected, then a female.
b. a female is selected, then a male.
c. two males are selected.
d. two females are selected.
e. no males are selected.
28. A math class consists of 25 students, 14 female and 11 male. Three students are selected at random to participate in a probability experiment. Compute the probability that
a. a male is selected, then two females.
b. a female is selected, then two males.
c. two females are selected, then one male.
d. three males are selected.
e. three females are selected.
29. A test was given to a group of students. The grades and gender are summarized below.
|
A |
B |
C |
Total |
|
|
Male |
8 |
18 |
13 |
39 |
|
Female |
10 |
4 |
12 |
26 |
|
Total |
18 |
22 |
25 |
65 |
Suppose a student is chosen at random:
a. Find the probability that the student is female and earned an A.
b. Find the probability that the student is male and earned an A.
c. Find the probability that the student is male or earned a \(\mathrm{C}\).
d. Find the probability that the student is female or earned a B.
30. The table below shows the number of credit cards owned by a group of individuals.
|
Zero |
One |
Two or more |
Total |
|
|
Male |
9 |
5 |
19 |
33 |
|
Female |
18 |
10 |
20 |
48 |
|
Total |
27 |
15 |
39 |
81 |
Suppose a person is chosen at random:
a. Find the probability that the person is male and has two or more credit cards.
b. Find the probability that the person is female and has one credit card.
c. Find the probability that the person is male or has no credit cards.
d. Find the probability that the person is female or has two or more credit cards.
31. Suppose we draw one card from a standard deck. What is the probability of drawing the King of hearts or a Queen?
32. Suppose you roll a six-sided die. What is the probability that you roll an even number or a five?
33. Suppose we draw one card from a standard deck. What is the probability of drawing a King or a heart?
34. Suppose we draw one card from a standard deck. What is the probability of drawing a face card or a diamond?
35. Suppose we draw one card from a standard deck. What is the probability of drawing a Jack or a black card?
36. Two six-sided dice are rolled. What is the probability of getting a sum of either 11 or 12 ?
37. A jar contains 6 red marbles numbered 1 to 6 and 8 blue marbles numbered 1 to 8 . A marble is drawn at random from the jar. Find the probability the marble is red or odd-numbered.
38. A jar contains 4 red marbles numbered 1 to 4 and 10 blue marbles numbered 1 to 10. A marble is drawn at random from the jar. Find the probability the marble is blue or even-numbered.
39. A jar contains 4 red marbles numbered 1 to 4 and 12 blue marbles numbered 1 to 12. A marble is drawn at random from the jar. Find the probability of the given event.
a. The marble is red or odd-numbered
b. The marble is blue and even-numbered
40. Given the following information, determine \(P(A\) or \(B)\).
\[
\begin{array}{l}
P(A)=0.56 \\
P(B)=0.53 \\
P(A \text { and } B)=0.42 \\
P(B \mid A)=0.3
\end{array}
\]
41. Given the following information, determine \(P(A\) or \(B)\).
\[
\begin{array}{l}
P(A)=0.76 \\
P(B)=0.3 \\
P(A \text { and } B)=0.28 \\
P(B \mid A)=0.2
\end{array}
\]