# Bayes' Theorem is given by P(A|B) = frac{P(B|A) cdot P(A)} {P(B)}. Use a two-way table to write an example of Bayes' Theorem.

Question
Two-way tables
Bayes' Theorem is given by P(A|B) = \frac{P(B|A) \cdot P(A)} {P(B)}. Use a two-way table to write an example of Bayes' Theorem.

2020-10-21
Possible answer: The two-way tables from exercise 7 was: $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{c}{\mid}{c}\right\rbrace}&{L}{e}{f}{t}&{R}{i}{g}{h}{t}&{T}{o}{t}{a}{l}\backslash{h}{l}\in{e}{F}{e}{m}{a}\le&{11}&{104}&{115}\backslash{h}{l}\in{e}{M}{a}\le&{24}&{92}&{116}\backslash{h}{l}\in{e}{T}{o}{t}{a}{l}&{35}&{196}&{231}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ Two events could then be male (A) and right handed (B). From the table, $$\displaystyle{P}{\left({A}\right)}={\frac{{{116}}}{{{31}}}}$$ and $$\displaystyle{P}{\left({B}\right)}={\frac{{{196}}}{{{231}}}}$$ since there are 116 males and 196 right handed people. Since there are 92 right handed males and 116 males, then $$\displaystyle{P}{\left({B}{\mid}{A}\right)}={\frac{{{92}}}{{{116}}}}$$ From the table, there are 196 right handed people and 92 of them are male so $$\displaystyle{P}{\left({A}{\mid}{B}\right)}={\frac{{{92}}}{{{196}}}}={\frac{{{23}}}{{{49}}}}.$$ Using Bayes’ Theorem, $$\displaystyle{P}{\left({B}{\mid}{A}\right)}={\frac{{{P}{\left({B}{\mid}{A}\right)}\cdot{P}{\left({A}\right)}}}{{{P}{\left({B}\right)}}}}={\frac{{{\frac{{{92}}}{{{116}}}}\cdot{\frac{{{116}}}{{{231}}}}}}{{{\frac{{{196}}}{{{231}}}}}}}$$ which is equal to the previous answer you found.

### Relevant Questions

Men and women were surveyed regarding their favorite leisure sport, as shown below. All questions pertain to this two-way frequency table.
$$\begin{array}{|c|c|c|}\hline\text{Leisure Sport}&\text{Golf}&\text{Tennis}&\text{Skiing}&\text{Total}\\\hline\text{Men} &32 & 21 & 30&83\\ \hline\text{Women}& 35 & 30&26&91\\\hline \ \text{Total}&67&51&56&174 \\ \hline \end{array}$$
$$\text{Find P(men)} \cdot \text{P(skiing)}.$$
(Choose a numbered choice from the list below.)
$$1) \frac{83}{174}$$
$$2) \frac{56}{174}4$$
$$3) \frac{4648}{174}$$
$$4) \frac{4648}{174^2}$$
The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P $$(A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).$$
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
How are the smoking habits of students related to their parents' smoking? Here is a two-way table from a survey of student s in eight Arizona high schools:
$$\begin{array}{c|c}&\text{Student smokes}&\text{Student does not smoke}&\text{Total}\\\hline\text{Both parents smoke}&400&1380&400+1380=1780\\\hline\text{One parent smokes}&416&1823&416+1823=2239\\\hline\text{Neither parent smokes}&188&1168&188+1168=1356\\\hline\text{Total}&400+416+188=1004&1380+1823+1168=4371&1004+4371=5375\end{array}$$
(a) Write the null and alternative hypotheses for the question of interest.
(b) Find the expected cell counts. Write a sentence that explains in simple language what "expected counts" are.
(c) Find the chi-square statistic, its degrees of freedom, and the P-value.
The following is a two-way table showing preferences for an award (A, B, C) by gender for the students sampled in survey. Test whether the data indicate there is some association between gender and preferred award.
$$\begin{array}{|c|c|c|}\hline &\text{A}&\text{B}&\text{C}&\text{Total}\\\hline \text{Female} &20&76&73&169\\ \hline \text{Male}&11&73&109&193 \\ \hline \text{Total}&31&149&182&360 \\ \hline \end{array}\\$$
Chi-square statistic=?
p-value=?
Conclusion: (reject or do not reject $$H_0$$)
Does the test indicate an association between gender and preferred award? (yes/no)
$$\begin{array}{|c|c|c|c|}\hline& \text{SUV} & \text{Sedan} & \text{Totals} \\ \hline\text{Male} & 21&39&60\\\hline\text{Female}&&45&180\\\hline\text{Total}&156&84&\\\hline\end{array}$$ The two-way table represents the results of a random survey taken to determine the preferred vehicle for male and female drivers. Given that the participant is a female, which choice is the conditional relative frequency that she prefers an SUV a)0.25 b)0.55 c)0.75 d)0.87
Is there a relationship between gender and relative finger length? To find out, we randomly selected 452 U.S. high school students who completed a survey. The two-way table summarizes the relationship between gender and which finger was longer on the left hand (index finger or ring finger).
$$\begin{array} {lc} & \text{Gender} \ \text {Longer finger} & \begin{array}{l|c|r|r} & \text { Female } & \text { Male } & \text { Total } \\\hline \text { Index finger } & 78 & 45 & 123 \\\hline \text{ Ring finger } & 82 & 152 & 234 \\ \hline \text { Same length } & 52 & 43 & 95 \\ \hline \text { Total } & 212 & 240 & 452 \end{array}\ \end{array}$$
Suppose we randomly select one of the survey respondents. Define events R: ring finger longer and F: female. Given that the chosen student does not have a longer ring finger, what's the probability that this person is male? Write your answer as a probability statement using correct symbols for the events.
In 1912 the Titanic struck an iceberg and sank on its first voyage. Some passengers got off the ship in lifeboats, but many died. The following two-way table gives information about adult passengers who survived and who died, by class of travel.
$$\begin{array} {lc} & \text{Class} \ \text {Survived } & \begin{array}{c|c|c|c} & \text { First } & \text { Second } & \text { Third } \\ \hline \text { Yes } & 197 & 94 & 151 \\ \hline \text { No } & 122 & 167 & 476 \end{array}\ \end{array}$$
Suppose we randomly select one of the adult passengers who rode on the Titanic. Define event D as getting a person who died and event F as getting a passenger in first class. Find P(not a passenger in first class or survived).
One hundred adults and children were randomly selected and asked whether they spoke more than one language fluently. The data were recorded in a two-way table. Maria and Brennan each used the data to make the tables of joint relative frequencies shown below, but their results are slightly different. The difference is shaded. Can you tell by looking at the tables which of them made an error? Explain.
$$\begin{array}{c|c}&Yes&No\\\hline\text{Children}&0.15&0.25\\\hline\text{Adults}&0.1&0.6\end{array}$$
A random sample of 2,500 people was selected, and the people were asked to give their favorite season. Their responses, along with their age group, are summarized in the two-way table below.
$$\begin{array}{c|cccc|c} & \text {Winter} &\text{Spring}& \text {Summer } & \text {Fall}& \text {Total}\\ \hline \text {Children} & 30 & 0 & 170&0&200 \\ \text{Teens} & 150 & 75 & 250&25&500 \\ \text {Adults } & 250 & 250 & 250&250&1000 \\ \text {Seniors} & 300 & 150 & 50&300&800 \\ \hline \text {Total} & 730 & 475 & 720 &575&2500 \end{array}$$
Among those whose favorite season is spring, what proportion are adults?
$$a) \frac{250}{1000}$$
$$b) \frac{250}{2500}$$
$$c) \frac{475}{2500}$$
$$d) \frac{250}{475}$$
$$e) \frac{225}{475}$$
$$\begin{array}{|c|c|c|}\hline &\text{Baseball}&\text{Basketball}&\text{Football}&\text{Hockey}&\text{Other}&\text{Total}\\\hline \text{Boys} &18&14&20&6&2&60\\ \hline \text{Girls}&14&16&13&5&12&60\\ \hline \text{Total}&32&30&33&11&14&120\\ \hline \end{array}\\$$