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A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Let T be the event of getting the white ball twice, F the event of picking the white ball first, S the event of picking the white ball in the second drawing.

  1. Compute P ( T ).
  2. Compute P ( T | F ).
  3. Are T and F independent?.
  4. Are F and S mutually exclusive?
  5. Are F and S independent?
  1. P ( T ) = 1 4
  2. P ( T | F ) = 1 2
  3. No
  4. No
  5. Yes
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Lopez, Shane, Preety Sidhu. “U.S. Teachers Love Their Lives, but Struggle in the Workplace.” Gallup Wellbeing, 2013. http://www.gallup.com/poll/161516/teachers-love-lives-struggle-workplace.aspx (accessed May 2, 2013).

Data from Gallup. Available online at www.gallup.com/ (accessed May 2, 2013).

Chapter review

Two events A and B are independent if the knowledge that one occurred does not affect the chance the other occurs. If two events are not independent, then we say that they are dependent.

In sampling with replacement, each member of a population is replaced after it is picked, so that member has the possibility of being chosen more than once, and the events are considered to be independent. In sampling without replacement, each member of a population may be chosen only once, and the events are considered not to be independent. When events do not share outcomes, they are mutually exclusive of each other.

Formula review

If A and B are independent, P ( A AND B ) = P ( A ) P ( B ), P ( A | B ) = P ( A ) and P ( B | A ) = P ( B ).

If A and B are mutually exclusive, P ( A OR B ) = P ( A ) + P ( B ) and P ( A AND B ) = 0.

E and F are mutually exclusive events. P ( E ) = 0.4; P ( F ) = 0.5. Find P ( E F ).

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J and K are independent events. P ( J | K ) = 0.3. Find P ( J ).

P ( J ) = 0.3

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U and V are mutually exclusive events. P ( U ) = 0.26; P ( V ) = 0.37. Find:

  1. P ( U AND V ) =
  2. P ( U | V ) =
  3. P ( U OR V ) =
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Q and R are independent events. P ( Q ) = 0.4 and P ( Q  AND  R ) = 0.1. Find P ( R ).

P ( Q AND R ) = P ( Q ) P ( R )

0.1 = (0.4) P ( R )

P ( R ) = 0.25

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Bringing it together

A previous year, the weights of the members of the San Francisco 49ers and the Dallas Cowboys were published in the San Jose Mercury News . The factual data are compiled into [link] .

Shirt# ≤ 210 211–250 251–290 290≤
1–33 21 5 0 0
34–66 6 18 7 4
66–99 6 12 22 5

For the following, suppose that you randomly select one player from the 49ers or Cowboys.

If having a shirt number from one to 33 and weighing at most 210 pounds were independent events, then what should be true about P (Shirt# 1–33|≤ 210 pounds)?

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The probability that a male develops some form of cancer in his lifetime is 0.4567. The probability that a male has at least one false positive test result (meaning the test comes back for cancer when the man does not have it) is 0.51. Some of the following questions do not have enough information for you to answer them. Write “not enough information” for those answers. Let C = a man develops cancer in his lifetime and P = man has at least one false positive.

  1. P ( C ) = ______
  2. P ( P | C ) = ______
  3. P ( P | C' ) = ______
  4. If a test comes up positive, based upon numerical values, can you assume that man has cancer? Justify numerically and explain why or why not.

  1. P ( C ) = 0.4567
  2. not enough information
  3. not enough information
  4. No, because over half (0.51) of men have at least one false positive text
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Given events G and H : P ( G ) = 0.43; P ( H ) = 0.26; P ( H AND G ) = 0.14

  1. Find P ( H OR G ).
  2. Find the probability of the complement of event ( H AND G ).
  3. Find the probability of the complement of event ( H OR G ).
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Given events J and K : P ( J ) = 0.18; P ( K ) = 0.37; P ( J OR K ) = 0.45

  1. Find P ( J AND K ).
  2. Find the probability of the complement of event ( J AND K ).
  3. Find the probability of the complement of event ( J AND K ).
  1. P ( J OR K ) = P ( J ) + P ( K ) − P ( J AND K ); 0.45 = 0.18 + 0.37 - P ( J AND K ); solve to find P ( J AND K ) = 0.10
  2. P (NOT ( J AND K )) = 1 - P ( J AND K ) = 1 - 0.10 = 0.90
  3. P (NOT ( J OR K )) = 1 - P ( J OR K ) = 1 - 0.45 = 0.55
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Questions & Answers

How to answer quantitative data
Alhassan Reply
what's up here ... am new here
sorry question a bit unclear...do you mean how do you analyze quantitative data? If yes, it depends on the specific question(s) you set in the beginning as well as on the data you collected. So the method of data analysis will be dependent on the data collecter and questions asked.
how to solve for degree of freedom
Quantitative data is the data in numeric form. For eg: Income of persons asked is 10,000. This data is quantitative data on the other hand data collected for either make or female is qualitative data.
Degree of freedom is the unconditionality. For example if you have total number of observations n, and you have to calculate variance, obviously you will need mean for that. Here mean is a condition, without which you cannot calculate variance. Therefore degree of freedom for variance will be n-1.
data that is best presented in categories like haircolor, food taste (good, bad, fair, terrible) constitutes qualitative data
vegetation types (grasslands, forests etc) qualitative data
I don't understand how you solved it can you teach me
Caleb Reply
solve what?
What is the end points of a confidence interval called?
lower and upper endpoints
Class members write down the average time (in hours, to the nearest half-hour) they sleep per night.
William Reply
how we make a classes of this(170.3,173.9,171.3,182.3,177.3,178.3,174.175.3)
why is always lower class bundry used
Assume you are in a class where quizzes are 20% of your grade, homework is 20%, exam _1 is 15%,exam _2 is 15%, and the final exam is 20%.Suppose you are in the fifth week and you just found out that you scored a 58/63 on the fist exam. You also know that you received 6/9,8/10,9/9 on the first
Diamatu Reply
quizzes as well as a 9/11,10/10,and 4.5/7 on the first three homework assignment. what is your current grade in the course?
the answer is 2.6
if putting y=3x examine that correlation coefficient between x and y=3x is 1.
Aadrsh Reply
what is permutation
Rodlett Reply
how to construct a histogram
Baalisi Reply
You have to plot the class midpoint and the frequency
ok so you use those two to draw the histogram right.
ok can i be a friend so you can be teaching me small small
how do you calculate cost effectiveness?
Hi everyone, this is a very good statistical group and am glad to be part of it. I'm just not sure how did I end up here cos this discussion just popes on my screen so if I wanna ask something in the future, how will I find you?
To make a histogram, follow these steps: On the vertical axis, place frequencies. Label this axis "Frequency". On the horizontal axis, place the lower value of each interval. ... Draw a bar extending from the lower value of each interval to the lower value of the next interval.
I really appreciate that
umar Reply
I want to test linear regression data such as maintenance fees vs house size. Can I use R square, F test to test the relationship? Is the good condition of R square greater than 0.5
Mok Reply
yes of course must have use f test and also use t test individually multple coefficients
hi frnd I'm akeem by name, I wanna study economics and statistics wat ar d thing I must do to b a great economist
Is R square cannot analysis linear regression of X vs Y relationship?
To be an economist you have to be professional in maths
hi frnds
what is random sampling what is sample error
Nistha Reply
@Nistha Kashyap Random sampling is the selection of random items (or random numbers) from the group. A sample error occurs when the selected samples do not truely represent the whole group. The can happen when most or all of the selected samples are taken from only one section of the group;
Thus the sample is not truely random.
What is zero sum game?
Hassan Reply
A game in which there is no profit & no loss to any of the both player.
Differences between sample mean & population mean
mohammed Reply
Not difference in the formula except the notation, sample mean is denoted by x bar and population mean is denoted by mu symbol. There is formula as well as notation between difference variance and standard deviations
Likely the difference would be in the result, unless the sample is an exact representation of the population (which is unlikely.)
what is data
Nii Avin - Data is just a simple way to refer to the numbers in the population, or in the sample used in your calculations.
what are the types of data
Data is the very pale android from the Star Trek Enterprise
Am Emmanuel from Nigeria
Am Qudus from Nigeria
am Handson from Cameroon
what is a mode?
Nii - data is whatever you are sampling. Such as the number of students in each classroom.
Handson Ndintek - the mode is the number appearing most frequently. Example: 7 9 11 7 4 6 3 7 2. 7 is the mode. In a group such as 7 9 1 4 6 3, there is no mode because no number appears more often than any other.
hi I want to know how to find class boundary
give me the two types of data
Neddy Reply
qualitative and quantitative
primary and secondary data
qualitative and quantitative
Using Cauchy Schwartz inequality,or prove that b2-b1-1=0
Md Reply
what is the ongoing probability that President Trump will remain in the position he has chosen as his viability of his cabinet as he runs for reelection in the primaries of 2020 election year

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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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