Question 1036314
Problem:


An investor invests in stocks without being informed of all the intricacies of stock market. The probability that his 1st investment in one market sector is a good stock is 1/3 and that his second investment in an unrelated market sector is a good stock is 1/4. 
Find THE probability that: 
A)At least one investment would be a good stock 
B)Both the investments would be good stocks
C) Only one investment would be a good stock 
D)None of the investments would be a good stock. 

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Steps: 

Let
X = event that his 1st investment is a good stock
X' = event that his 1st investment is NOT a good stock
Y = event that his 2nd investment is a good stock
Y' = event that his 2nd investment is NOT a good stock


So the corresponding probabilities are
P(X) = 1/3
P(Y) = 1/4
P(X') = 1 - (1/3) = 3/3 - 1/3 = 2/3
P(Y') = 1 - (1/4) = 4/4 - 1/4 = 3/4


In short,
P(X) = 1/3
P(Y) = 1/4
P(X') = 2/3
P(Y') = 3/4


Let's move onto part (A)


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(A)

The two events are independent, so we can use the rule that P(X and Y) = P(X)*P(Y)


The probability that BOTH investments are bad investments is the same as asking for the probability of event X' happening and event Y' happening as well.


P(X' and Y') = P(X')*P(Y')
P(X' and Y') = (2/3)*(3/4)
P(X' and Y') = (2*3)/(3*4)
P(X' and Y') = 6/12
P(X' and Y') = 1/2


Subtract this from 1


1 - (probability both bad) = 1 - (1/2) = 2/2 - 1/2 = 1/2


So the probability of having at least one good investment is 1/2


The final answer to part (A) is <font color=red>1/2</font>

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(B)

The events X and Y are independent so P(X and Y) = P(X)*P(Y)

P(X and Y) = P(X)*P(Y)
P(X and Y) = (1/3)*(1/4)
P(X and Y) = (1*1)/(3*4)
P(X and Y) = 1/12


The final answer to part B is <font color=red>1/12</font>

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(C)


Add the probabilities P(X) and P(Y). Then subtract off the probability found in part (B) to get the final answer


P(X or Y) = P(X) + P(Y) - P(X and Y)


Note: when I say P(X or Y) I mean you can only pick one event (X or Y). You cannot pick both. This is an exclusive 'or'.



So using that formula gives us this


P(X or Y) = P(X) + P(Y) - P(X and Y)
P(X or Y) = 1/3+1/4 - 1/12
P(X or Y) = 4/12+3/12 - 1/12
P(X or Y) = (4+3-1)/12
P(X or Y) = 6/12
P(X or Y) = 1/2



Answer to part C is <font color=red>1/2</font>


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(D)


"None of the investments would be a good stock" is the same as "Both investments are bad"


P(X' and Y') = P(X')*P(Y')
P(X' and Y') = (2/3)*(3/4)
P(X' and Y') = (2*3)/(3*4)
P(X' and Y') = 6/12
P(X' and Y') = 1/2



Answer to part D is <font color=red>1/2</font>


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Summary:


Answer to part A: <font color=red>1/2</font>
Answer to part B: <font color=red>1/12</font>
Answer to part C: <font color=red>1/2</font>
Answer to part D: <font color=red>1/2</font>


Answers are given in fraction form.