4.3.1: Exercises

1. Compute the probability of rolling a 12-sided die and getting a number other than 8.
2. If you pick one card at random from a standard deck of cards, what is the probability it is not the Ace of Spades?
3. Referring to the grade table below, what is the probability that a student chosen at random did NOT earn a C?

   \(\begin{array}{|l|l|l|l|l|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Total } \\ \hline \text { Male } & 8 & 18 & 13 & 39 \\ \hline \text { Female } & 10 & 4 & 12 & 26 \\ \hline \text { Total } & 18 & 22 & 25 & 65 \\ \hline \end{array}\)

4. Referring to the credit card table below, what is the probability that a person chosen at random has at least one credit card?

   \(\begin{array}{|l|l|l|l|l|} \hline & \text { Zero } & \text { One } & \text { Two or more } & \text { Total } \\ \hline \text { Male } & 9 & 5 & 19 & 33 \\ \hline \text { Female } & 18 & 10 & 20 & 48 \\ \hline \text { Total } & 27 & 15 & 39 & 81 \\ \hline \end{array}\)

5. A six-sided die is rolled twice. What is the probability of showing a 6 on both rolls?
6. A fair coin is flipped twice. What is the probability of showing heads on both flips?
7. A six-sided die is rolled twice. What is the probability of showing a 5 on the first roll and an even number on the second roll?
8. Suppose that 21% of people own dogs. If you pick two people at random, what is the probability that they both own a dog?
9. Suppose a jar contains 17 red marbles and 32 blue marbles. If you reach in the jar and pull out 2 marbles at random without replacement, find the probability that both are red.
10. 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.
11. 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:
    1. Bert and Ernie both draw an Ace.
    2. Bert draws an Ace but Ernie does not.
    3. neither Bert nor Ernie draws an Ace.
    4. Bert and Ernie both draw a heart.
    5. Bert gets a card that is not a Jack and Ernie draws a card that is not a heart.
12. 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:
    1. Bert gets a Jack and Ernie rolls a five.
    2. Bert gets a heart and Ernie rolls a number less than six.
    3. Bert gets a face card (Jack, Queen or King) and Ernie rolls an even number.
    4. Bert gets a red card and Ernie rolls a fifteen.
    5. Bert gets a card that is not a Jack and Ernie rolls a number that is not twelve.
13. Compute the probability of drawing a King from a deck of cards and then drawing a Queen without replacement.
14. Compute the probability of drawing two spades from a deck of cards without replacement.
15. A math class consists of 25 students, 14 freshmen and 11 sophomores. Two students are selected at random to participate in a probability experiment. Compute the probability that
    1. a sophomore is selected, then a freshman.
    2. a freshman is selected, then a sophomore.
    3. two sophomores are selected.
    4. two freshmen are selected.
    5. no sophomores are selected.
16. A math class consists of 25 students, 14 freshmen and 11 sophomores. Three students are selected at random to participate in a probability experiment. Compute the probability that
    1. a sophomore is selected, then two freshmen.
    2. a freshmen is selected, then two sophomores.
    3. two freshmen are selected, then one sophomore.
    4. three sophomores are selected.
    5. three freshmen are selected.
17. Giving a test to a group of students, the grades and class are summarized below. If one student was chosen at random, find the probability that the student was a freshman and earned an A.

    \(\begin{array}{|l|l|l|l|l|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Total } \\ \hline \text { Sophomore } & 8 & 18 & 13 & 39 \\ \hline \text { Freshman } & 10 & 4 & 12 & 26 \\ \hline \text { Total } & 18 & 22 & 25 & 65 \\ \hline \end{array}\)

18. The table below shows the number of credit cards owned by a group of individuals. If one person was chosen at random, find the probability that the person was male and had two or more credit cards.

    \(\begin{array}{|l|l|l|l|l|} \hline & \text { Zero } & \text { One } & \text { Two or more } & \text { Total } \\ \hline \text { Male } & 9 & 5 & 19 & 33 \\ \hline \text { Female } & 18 & 10 & 20 & 48 \\ \hline \text { Total } & 27 & 15 & 39 & 81 \\ \hline \end{array}\)

19. 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.
20. 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.
21. Referring to the grade table below, find the probability that a student chosen at random is a freshman or earned a B.

    \(\begin{array}{|l|l|l|l|l|} \hline & \mathrm{A} & \mathrm{B} & \mathrm{C} & \text { Total } \\ \hline \text { Male } & 8 & 18 & 13 & 39 \\ \hline \text { Female } & 10 & 4 & 12 & 26 \\ \hline \text { Total } & 18 & 22 & 25 & 65 \\ \hline \end{array}\)

22. Referring to the table below, find the probability that a person chosen at random is male or has no credit cards.

    \(\begin{array}{|l|l|l|l|l|} \hline & \text { Zero } & \text { One } & \text { Two or more } & \text { Total } \\ \hline \text { Male } & 9 & 5 & 19 & 33 \\ \hline \text { Female } & 18 & 10 & 20 & 48 \\ \hline \text { Total } & 27 & 15 & 39 & 81 \\ \hline \end{array}\)

23. Compute the probability of drawing the King of hearts or a Queen from a deck of cards.
24. Compute the probability of drawing a King or a heart from a deck of cards.
25. 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
    1. Even-numbered given that the marble is red.
    2. Red given that the marble is even-numbered.
26. 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
    1. Odd-numbered given that the marble is blue.
    2. Blue given that the marble is odd-numbered.
27. Compute the probability of flipping a coin and getting heads, given that the previous flip was tails.
28. Find the probability of rolling a “1” on a fair die, given that the last 3 rolls were all ones.
29. Suppose a math class contains 25 students, 14 females (three of whom speak French) and 11 males (two of whom speak French). Compute the probability that a randomly selected student speaks French, given that the student is female.
30. Suppose a math class contains 25 students, 14 females (three of whom speak French) and 11 males (two of whom speak French). Compute the probability that a randomly selected student is male, given that the student speaks French.
31. 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 B be the event "the person tests positive".
    1. Find the probability that a person has the virus given that they have tested positive, i.e. find \(P(A | B)\).
    2. Find the probability that a person does not have the virus given that they test negative, i.e. find \(P(\text { not } A | \text { not } B)\).
32. 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 B be the event "the person tests positive".
    1. Find the probability that a person has the virus given that they have tested positive, i.e. find \(P(A | B)\).
    2. Find the probability that a person does not have the virus given that they test negative, i.e. find \(P(\text { not } A | \text { not } B)\).