Discrete Mathematics Probability Worksheet Name __________________________________________

(1) Two ordinary dice are rolled. Find the probability that ...

(a) ... the sum of the dice is 6, 7 or 8.

(b) ... the sum of the dice is 5 or at least one of the dice shows a 5

.(c) ... the two dice match.

(2) A card is drawn from an ordinary deck of 52 cards. Find the probability that the card is ...

(a) ... an ace or a heart

.(b) ... an ace or a black card

.(c) ... a diamond, a club, or a king.

(3) Two cards are drawn from deck, with replacement. (This means that one person chooses a card,looks at it and returns it, and then another person chooses a card, looks at it, and returns it.) What is the probability that ...

(a) ... the first card is an ace and the second card is black?

(b) ... both cards are spades?

(c) ... neither card has a value from {2, 3, 4, 5}?(d) ... at least one card is an ace?

(e) ... the first card is an ace or the second card is black?

(4) An urn contains 7 red marbles labeled {1,2,3,4,5,6,7} and 5 green marbles labeled {1,2,3,4,5}.Four marbles are pulled out at once (i.e. with no particular order). What is the probability that ...

(a) ... all four marbles are red?

(b) ... more of the marbles are green than red?

(c) ... both red and green marbles are present?

(d) ... two of the marbles chosen are both labeled "5"?

(5) What is the probability that a five card hand dealt from a standard deck of cards will include fourcards of the same value? (This kind of hand is called a "four of a kind" in Poker.)

(6) A fair coin is tossed ten times in a row.

(a) What is the probability that "heads" comes up exactly five times?

(b) What is the probability that "heads" come up at least eight times?

(c) What is the probability that "heads" come up at least once?

You flip a coin 8 times. What is the probability of seeing exactly four tails?

128/256

186/256

70/256

4/256

(7) Let's say the probability of having a particular cancer is 1%.There is a test for this cancer. It will test positive 90% of the time if you have the cancer and it will correctly come out negative 80% of the time if you don't have the cancer.

Fill out the following probabilities:

Pr(cancer) = 0.01

Pr(no cancer) =

Pr(positive | cancer) = 0.9

Pr(negative | cancer) =

Pr(positive | no cancer) =

Pr(negative | no cancer) = 0.8

(8) By the product rule, the probability of the test coming out positive and you have cancer is:

A survey was given to 18 students. One question asked about the one-way distance the student had to travel to attend college. The results, in miles, are shown in the following table. Use the median procedure for finding quartiles to find the first, second, and third quartiles for the data. Distance Traveled to Attend College

46 50 18 26 64 78 4 38 44 44 10 70 74 44 86 32 26 48

Q1 =

Q2 =

Q3 =

The algae dataset (last page) uses repeated measures: chlorophyll concentrations are measured at three depths in each of 4 lakes. Fit two models to test for an effect of depth on chlorophyll concentration: one with only depth as a predictor, and one with both depth as the predictor and lake as the “subject”. Produce ANOVA tables for both models. Create a graph showing how chlorophyll concentration changes with depth. Describe each result, then compare them and explain any differences.

Before you do any of this, make sure your depth variable is being treated as a categorical variable, with a logical order for the values.

algae

An open empty tank (filled with atmospheric air at 27oC and 100 kPa) is filled with liquid nitrogen as much as 14 kg quickly. After that, the tank is immediately closed and allowed to stand for a certain period of time. The tank is equipped with an open pipe manometer where the manometer fluid used is mercury (s.g. = 13.55). In the condition of the closed tank, the results of reading the manometer when it is constant shows that the difference in mercury surface height is 38 cm.

If it is assumed:
1. This system runs isothermal.
2. Nitrogen gas and air follow the ideal gas principle.
3. During the tank closing process, no nitrogen gas is lost.
4. Because liquid nitrogen intake and tank closure are fast, the volume of liquid nitrogen can be neglected meaning that at t = 0, the tank is considered to only contain air.

Therefore,
a. Determine the tank volume used!
b. Determine the number of moles of air in the tank!
c. Determine the specific gravity value of the gas mixture!

How much time do you spend talking on your phone per day? Initial post by Wednesday at 11:59 PM: Guess the number of minutes you think you spend talking on the phone each day. This is your null hypothesis. Posting from Thursday to Sunday, 11:59 PM: Secure the data from your phone for the past month and conduct a one sample hypothesis test of the mean length of your phone calls per day. Be sure to show all your work by using excel and include your data set. Please check out the video if you need help on using excel to find the mean and sample standard deviation. Video on using excel Use a significance level of 0.05. You will have 30 data values showing the total of minutes you spend per day. (The easiest way to secure this data is to look at your "Recent" calls on your phone and list the total number of minutes that you have spent on the phone each day in the last 30 days. This will be the data that you will use.)

My guess is 60 mins per day, below is the data to utilize........

Sign Runs Test/G

Consider the following sequence of 1 and 6:

6 111 66 111 66 11 6 11 666 11 6 1111 666 11 66

With a 0.01 significance level, we wish to test the claim that the above sequence was produced in a random manner. Answer each of the following questions

(a) The null hypothesis H0 is given by

A. n1=n2
B. ρ=0
C. β=0
D. The data are in an order that is not random
E. The data are in a random order
F. Median=0
G. G=0
H. r=0
I. None of the above.

(b) The null hypothesis H1 is given by

A. ρ≠0
B. G≠0
C. n1≠n2
D. The data are in an order that is not random
E. The data are in a random order
F. r≠0
G. Median ≠0
H. β≠0
I. None of the above.

(c) The number of runs G is

A. 27
B. 30
C. 15
D. 17
E. 21
F. 19
G. None of the above.

(d) What kind of test should you conduct

A. Both sign and goodness of fit tests
B. Sign test
C. Independence Test
D. Goodness of fit test
E. Runs test for randomness and the test statisc is G
F. Runs test for randomness and the test statistic is a z-score
G. None of the above.

(e) What is/are the critical(s) value(s)

A. The smallest one is -1.96 and the largest one is 1.96
B. The smallest one is 11 and the largest one is 24
C. The only critical value is 11
D. The negative one is -1.645 and the positive one is 1.645
E. The smallest one is -2.575 and the largest one is 2.575
F. The only critical value is 24
G. The only critical value is 23
H. The only critical value is -2.575
I. None of the above.

(f) The test statistic is

A. z=1.65 with n1=15 and n2=18.
B. G=15 with n1=15 and n2=18.
C. z=1.49 with μG=19.37 and σG=4.80.
D. z=1.56 with n1=18 and n2=15.
E. z=−.84 with μG=17.36 and σG=2.8.
F. z=1.56 with n1=15 and n2=18.
G. z=−.49 with μG=19.37 and σG=2.8.
H. G=15 with n1=18 and n2=15.
I. z=−.49 with G=17.36 and σG=4.80.
J. None of the above.

(g) The conclusion:

A. We reject H0 and then there is enough evidence to support the calim that the above sequence was in a random order
B. We fail to reject H0 and then there is enough evidence to support the calim that the above sequence was in a random order
C. We reject H0 and then there isn't enough evidence to support the calim that the above sequence was in a random order
D. We fail to reject H0 and then there isn't enough evidence to reject the calim that the above sequence was in a random order.
E. We fail to reject H0 and then there isn't enough evidence to support the calim that the above sequence was in a random order
F. We fail to reject H0 and then there is enough evidence to reject the calim that the above sequence was in a random order
G. We reject H0 and then there is enough evidence to reject the calim that the above sequence was in a random order
H. We reject H0 and then there isn't enough evidence to reject the calim that the above sequence was in a random order
I. None of the above.

The graph of a function

y = g(x)

on the domain

−8 ≤ x ≤ 8

consists of line segments and semicircles of radius 2 connecting the points

(−8, 0), (−4, 4), (0, 4), (4, 4), (8, 0).

(a) What is the range of g?

0 < y < 60 ≤ y ≤ 4    0 ≤ y ≤ 20 < y < 40 ≤ y ≤ 6

(b) Where is the function increasing? (Select all that apply.)

−8 ≤ x ≤ −2−2 ≤ x ≤ 22 ≤ x ≤ 44 ≤ x ≤ 8−8 ≤ x ≤ 8

Where is the function decreasing? (Select all that apply.)

−8 ≤ x ≤ −2−2 ≤ x ≤ 22 ≤ x ≤ 44 ≤ x ≤ 8−8 ≤ x ≤ 8

(c) Find the multipart formula for y = g(x) if

(d) If we restrict the function to the smaller domain

−6 ≤ x ≤ 0,

what is the range?

0 ≤ y ≤ 62 ≤ y ≤ 6    0 ≤ y ≤ 42 ≤ y ≤ 40 ≤ y ≤ 2

(e) If we restrict the function to the smaller domain

0 ≤ x ≤ 4,

what is the range?

2 ≤ y ≤ 40 ≤ y ≤ 2    0 ≤ y ≤ 64 ≤ y ≤ 60 ≤ y ≤ 4

Howard company makes two products A & B. Here is some financial information about those products.

7. A farmer wishes to determine if a new fertilizer increases her tomato crop yield. She lays out seven plots and in each plot plants two tomato plants. One plant is given the new fertilizer and the other is given the fertilizer the farmer is currently using. The table below has the yield (in pounds).

Plot 1 2 3 4 5 6 7
New    5.3       6.1        5.7        8.4       3.7        9.4        9.2
Old     4.9        5.9        6.0        6.9       2.8        8.2        7.2

Make a 95% confidence interval for the difference between the mean yield of all plants with the new fertilizer and with the old fertilizer. Interpret the interval.