Question 1202484
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Answer: No, the events are NOT independent.
The events are dependent somehow.


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Explanation:


Data table
<table border = "1" cellpadding = "5"><tr><td></td><td>in favor</td><td>against</td><td>total</td></tr><tr><td>male</td><td>14</td><td>46</td><td>60</td></tr><tr><td>female</td><td>5</td><td>35</td><td>40</td></tr><tr><td>total</td><td>19</td><td>81</td><td>100</td></tr></table>


Define these two events
F = a female employee is selected
H = the person is in favor of high salaries


There are at least two methods we can use to check if events F and H are independent or not.


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Method 1


If we can show P(F and H) = P(F)*P(H) is the case, then events F and H are independent.


According to the table
P(F) = (40 women)/(100 employees) = 40/100 = 2/5
P(H) = (19 in favor)/(100 employees) = 19/100
Then
P(F)*P(H) = (2/5)*(19/100) = 38/500 = 19/250


In the table there are 5 women in favor of higher salaries. Therefore P(F and H) = 5/100 = 1/20


Are the two fractions 19/250 and 1/20 equal?
Let's cross multiply to find out
19/250 = 1/20
19*20 = 250*1
380 = 250
The last statement is false, which must mean the first statement is false. The fractions 19/250 and 1/20 are not equivalent.


We could also use a calculator to get:
19/250 = 0.076
1/20 = 0.05
To help better see the results aren't the same.


Therefore P(F and H) = P(F)*P(H) is false.


Events F and H are NOT independent.
We consider them dependent.
F depends on H, or H depends on F. 
The two events are linked somehow.


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Method 2


Events F and H are independent if and only if:
P(F given H) = P(F)
P(H given F) = P(H)


If the events were independent, then the prior "given" shouldn't affect the probability.



As previously mentioned, back in method 1, the table says:
P(F) = 2/5
P(H) = 19/100


Let's compute P(F given H).
Because we know event H has occurred, we know 100% the person is in favor of high salaries.
That means we'll narrow our focus to the "in favor" column only.


There are 5 women who favor high salaries out of 19 people total who favor high salaries.
P(F given H) = 5/19
An alternative is to use this formula
P(F given H) = P(F and H)/P(H)
I'll let the student do this part.


Compare these two items
P(F) = 2/5
P(F given H) = 5/19
Those fractions are not equal (use the cross multiplication technique, or a calculator)
This means P(F given H) = P(F) is false. The two events are dependent.


Through similar logic, 
P(H given F) = 5/40 = 1/8
P(H) = 19/100


Use the technique shown in the previous method to conclude 19/100 = 1/8 is false, so P(H given F) = P(H) is also false.
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