Question 1200015: 1. The probability that a salmon swims successfully through a dam is 0.85. Find
the probability that two salmon swim successfully through the dam.
2. Two cards are selected from a standard deck without replacement. Find the
probability that they are both hearts.
a. Decide if the events are independent or dependent.
b. Use the Multiplication Rule to find the probability.
Answer by math_tutor2020(3817)   (Show Source):
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Problem 1
The events are Independent 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) = 0.7225 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 Dependent
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) = 1/17 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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