Question 1200015
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Problem 1


The events are <font color=red><u>&nbsp;&nbsp;Independent&nbsp;&nbsp;</u></font> since each salmon is separate from any other. 
One does not affect another. Each has probability of 0.85


A = salmon #1 makes it through
B = salmon #2 makes it through
P(A) = 0.85
P(B) = 0.85
P(A and B) = P(A)*P(B) .... since events are independent
P(A and B) = 0.85*0.85
P(A and B) = <font color=red>0.7225</font> is the exact probability that both salmon make it through.


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Problem 2


The key phrasing to look for here is "without replacement". 
This leads to the events being <font color=red><u>&nbsp;&nbsp;Dependent&nbsp;&nbsp;</u></font>


The first card selected is not put back. Nor is a copy put in its place. 
Therefore, the second selection's probability will be altered depending on what happens with the first selection. 


A = 1st selection is a heart
B = 2nd selection is a heart


There are 13 hearts out of 52 cards total.
P(A) = 13/52 = 1/4
P(B given A) = (12 hearts left)/(51 cards left) = 12/51 = 4/17


P(A and B) = P(A)*P(B given A)
P(A and B) = (1/4)*(4/17)
P(A and B) = <font color=red>1/17</font> is the probability both cards are hearts, assuming the first card is not put back (aka no replacement)


If events A and B were independent, then the P(B given A) can be replaced with P(B).
In other words,
P(B given A) = P(B) if and only if A & B are independent.
Also,
P(A given B) = P(A) if and only if A & B are independent.
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