Question 1172449
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Original Table
<table border = "1" cellpadding = "5"><tr><td></td><td>B</td><td>B'</td></tr><tr><td>A</td><td>30</td><td>40</td></tr><tr><td>A'</td><td>40</td><td>50</td></tr></table>
Compute the subtotals and the grand total
<table border = "1" cellpadding = "5"><tr><td></td><td>B</td><td>B'</td><td>Total</td></tr><tr><td>A</td><td>30</td><td>40</td><td>70</td></tr><tr><td>A'</td><td>40</td><td>50</td><td>90</td></tr><tr><td>Total</td><td>70</td><td>90</td><td>160</td></tr></table>
Divide each item by the grand total 160, and fully reduce, to compute the probabilities
<table border = "1" cellpadding = "5"><tr><td></td><td>B</td><td>B'</td><td>Total</td></tr><tr><td>A</td><td>3/16</td><td>1/4</td><td>7/16</td></tr><tr><td>A'</td><td>1/4</td><td>5/16</td><td>9/16</td></tr><tr><td>Total</td><td>7/16</td><td>9/16</td><td>1</td></tr></table>
We'll use this probability table to answer parts (a) through (d)


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Part (a)


P(A and B) = 3/16 .... upper left corner of table
P(B) = 7/16 .... bottom of column 1


P(A | B) = Probability of A, given B
P(A | B) = P(A and B)/P(B)
P(A | B) = (3/16)/(7/16)
P(A | B) = (3/16)*(16/7)
P(A | B) = 3/7


<font color=red>Answer: 3/7</font>
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Part (b)

P(A and B') = 1/4
P(B') = 9/16


P(A given B') = P(A and B')/P(B')
P(A given B') = (1/4) divide by (9/16)
P(A given B') = (1/4)*(16/9)
P(A given B') = (1*16)/(4*9)
P(A given B') = 16/36
P(A given B') = (4*4)/(4*9)
P(A given B') = 4/9


<font color=red>Answer: 4/9</font>


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Part (c)


P(A' and B') = 5/16
P(B') = 9/16


P(A' given B') = P(A' and B')/P(B')
P(A' given B') = (5/16) divide by (9/16)
P(A' given B') = (5/16)*(16/9)
P(A' given B') = 5/9


<font color=red>Answer: 5/9</font>
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Part (d)


Events A and B would be independent if and only if the following two items are true
P(A given B) = P(A)
P(B given A) = P(B)
Independent events are not linked together. If A and B are independent, then one event occurring does not change the probability of the other.


From part (a), we found P(A given B) = 3/7, but this is not the same value as P(A) = 7/16, which is what the table shows. This concludes that A and B are not independent.


As an alternative, we could also use the equation
P(A and B) = P(A)*P(B)
to find that
P(A and B) = P(A)*P(B)
3/16 = (7/16)*(7/16)
3/16 = 49/256
which is a false equation, so events A and B are not independent.


<font color=red>Answer: A and B are <u>not</u> independent. They are dependent.</font></font>