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Events A and B occur independently of one another.

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Events A and B occur independently of one another. [#permalink] New post   09 Feb 2021, 11:12
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Events A and B occur independently of one another. The probability that neither event occurs is 6 times the probability that both events occur. Furthermore, the probability that event B occurs and event A does not occur is 50 percent greater than the probability that event A occurs but event B does not occur. What is the probability that event B occurs?

A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4
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C
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Events A and B occur independently of one another. [#permalink] New post   22 Apr 2021, 00:47
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twobagels wrote:
Events A and B occur independently of one another. The probability that neither event occurs is 6 times the probability that both events occur. Furthermore, the probability that event B occurs and event A does not occur is 50 percent greater than the probability that event A occurs but event B does not occur. What is the probability that event B occurs?

A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4



Let the probability that event A and B occur be A and B respectively, so probability that A and B do not occur are (1-A) and (1-B) respectively.

Quote:
The probability that neither event occurs is 6 times the probability that both events occur.

The probability that neither event occurs = (1-A)(1-B)
The probability that both events occur = A*B
So (1-A)(1-B)=6AB......1-A-B+AB=6AB
Since we are looking at value of B, let us get the equation in terms of B lone.
1-B=5AB+A.......\(A=\frac{1-B}{5B+1}\)...(i)

Quote:
Furthermore, the probability that event B occurs and event A does not occur is 50 percent greater than the probability that event A occurs but event B does not occur.

The probability that event B occurs and event A does not occur = B*(1-A)
The probability that event A occurs but event B does not occur = A(1-B)
So, \(B(1-A)=\frac{3}{2}*A(1-B)\)..........
\(2B-2AB=3A-3AB.......2B=3A-AB\).......
A=\(\frac{2B}{3-B}\)...(ii)

Equating (i) and (ii)
\(A=\frac{1-B}{5B+1}=\frac{2B}{3-B}\).........
\((1-B)(3-B)=2B(5B+1)\).........
\(3-4B+B^2=10B^2+2B\).......
\(9B^2+6B-3=0.........3B^2+2B-1=0\)
\(3B^2+3B-B-1=0.......3B(B+1)-1(B+1)=0.......(3B-1)(B+1)=0\)

So B=-1 or B=\(\frac{1}{3}\)

As B cannot be less than 0, B=\(\frac{1}{3}\)

C
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Events A and B occur independently of one another. [#permalink] New post   20 Apr 2021, 06:31
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sthahvi

Fantastic and hard Algebra question.


Let the:

Prob. A occurs = A

Prob. A NOT occurs = X

Prob. B occurs = B

Prob. B NOT occurs = Y

Events A and B occur independently of each other. If we want to find the Probability that two independent events occur together, we multiply the respective Probabilities.

“Prob that neither event occurs is equal to 6 times the probability that both events occur”

(Prob of NOT A) * (Prob of NOT B) = (6) * (Prob of A) * (Prob of B)

XY = (6)AB

“Prob that event B occurs AND event A does NOT occur ———is 50% greater than ———the Prob that event A occurs AND Prob that event B does NOT occur.”

Again, since the events of one event occurring and the other event not occurring are Indpendent Events we can multiply the individual probabilities to find the Prob that they occur together (“AND” Probability)

BX = (1.5)AY

Since each variable represents a positive probability value, we can divide the 1st equation above by this equation


XY = (6)AB

___________

BX = (1.5)AY


The X and A cancels out in the NUM/DEN

We are left with the equation:

Y/B = (6B) / (1.5Y)


1.5 divides evenly into 6 four times

Y/B = (4B) / Y

Cross multiply

Y^2 = (4) (B)^2

Again, since we are dealing with positive probabilities, when we take the even root (square root) we do not have to worry about the negative root

Take the square root of both sides

Y = 2B

Or in Ratio Form:

Y/B = 2/1

B ——- stands for he Prob that event B DOES occur

Y ———-stands for the Prob that event B does NOT occur

Together, B and Y are completely exhaustive of the possibilities. In other words:

Prob of B occurring + Prob of B NOT occurring = 1

Y/B = 2/1

Is he same as

Y : B = 2 : 1

Y (B not occurring) is twice as likely to occur as B is

Which means out of the complete sample space, the Prob that B will occur = 1/3

Or, using algebra


Y + B = 1

Y = 2B

2B + B = 1 ———-> B = 1/3

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Re: Events A and B occur independently of one another. [#permalink] New post   10 Feb 2021, 19:21
Does someone have the answer for this?

I couldn't quite solve it even using a table approach

P(A and B) = x
P(B and Not A) = 1.5y
P(A and Not B) = y
P(Neither) = 6x

Got stuck after.
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Re: Events A and B occur independently of one another. [#permalink] New post   11 Feb 2021, 05:11
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can someone explain the answer of this
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Events A and B occur independently of one another. [#permalink] New post   21 Apr 2021, 21:47
Here is my approach
Remember: Events A and B are independent then P(A∩B) = P(A) · P(B)
Note: P(A) is the probability of event A occur
P(A') is the probability of event A NOT occur (1-P(A)
P(B) is the probability of event B occur
P(B') is the probability of event B NOT occur (1-P(B)
Having: A and B occur independently of one another (A occur not depend on B)=> P(A and B) = P(A)*P(B)
=> A' and B' occur independently (or P(A' and B') = P(A')*P(B'))
A and B' occur independently (or P(A and B') = P(A)*P(B'))
B and A' occur independently (or P(B and A') = P(B)*P(A'))
On the question:
1. The probability that neither event occurs (P(A' and B')) is 6 times the probability that both events occur(P(A and B)).
\(=> P(A' and B') = 6*P(A and B)\\
<=>P(A')*P(B') = 6*P(A)*P(B) (1)\)
2. the probability that event B occurs and event A does not occur(P(B and A')) is 50 percent greater than the probability that event A occurs but event B does not occur (P(A and B'))
=> P(B and A') is 50% greater than P(A and B')
So if P(A and B') is 100% then P(B and A') is 100%+50% = 150% P(A and B')
=> P(B and A') = 150% P(A and B')
<=> P(B)*P(A') = 150%P(A)*P(B') (2)
From (1) (2), divide (1) by (2)
\(\frac{P(A')*P(B')}{(P(B)*P(A')} = \frac{6*P(A)*P(B)}{(150%*P(A)*P(B')}\)
\(<=> \frac{P(B')}{P(B)} = \frac{4*P(B)}{P(B')}\)
\(<=>\frac{P(B')^2}{P(B)^2} = 4\)
\(<=>\frac{P(B')}{P(B)} = 2\)
<=> P(B') = 2P(B)
Having P(B)+P(B') = 1
=> P(B)+2P(B) = 1
=> 3P(B) = 1
=> P(B) = 1/3
Hence answer C
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Re: Events A and B occur independently of one another. [#permalink] New post   11 Jun 2021, 10:12
chetan2u wrote:
twobagels wrote:
Events A and B occur independently of one another. The probability that neither event occurs is 6 times the probability that both events occur. Furthermore, the probability that event B occurs and event A does not occur is 50 percent greater than the probability that event A occurs but event B does not occur. What is the probability that event B occurs?

A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4



Let the probability that event A and B occur be A and B respectively, so probability that A and B do not occur are (1-A) and (1-B) respectively.

Quote:
\(3B^2+3B-B-1=0.......3B(B+1)-1(B+1)=0.......(3B-1)(B+1)=0\)

So B=-1 or B=\(\frac{1}{3}\)

As B cannot be less than 0, B=\(\frac{1}{3}\)


C


The coeff of x^2 needs to be 1 for factorizing like that.
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Re: Events A and B occur independently of one another. [#permalink] New post   11 Jun 2021, 10:37
Expert Reply
garima2709 wrote:
chetan2u wrote:
twobagels wrote:
Events A and B occur independently of one another. The probability that neither event occurs is 6 times the probability that both events occur. Furthermore, the probability that event B occurs and event A does not occur is 50 percent greater than the probability that event A occurs but event B does not occur. What is the probability that event B occurs?

A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4



Let the probability that event A and B occur be A and B respectively, so probability that A and B do not occur are (1-A) and (1-B) respectively.

Quote:
\(3B^2+3B-B-1=0.......3B(B+1)-1(B+1)=0.......(3B-1)(B+1)=0\)

So B=-1 or B=\(\frac{1}{3}\)

As B cannot be less than 0, B=\(\frac{1}{3}\)


C


The coeff of x^2 needs to be 1 for factorizing like that.



No, it need not be.

You have to multiply the coefficient of x^2 and the constant( the term without x), and try to get the coefficient of x in terms of two factors of that product.
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Re: Events A and B occur independently of one another. [#permalink] New post   12 Jun 2022, 22:49
chetan2u wrote:
twobagels wrote:
Events A and B occur independently of one another. The probability that neither event occurs is 6 times the probability that both events occur. Furthermore, the probability that event B occurs and event A does not occur is 50 percent greater than the probability that event A occurs but event B does not occur. What is the probability that event B occurs?

A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4



Let the probability that event A and B occur be A and B respectively, so probability that A and B do not occur are (1-A) and (1-B) respectively.

Quote:
The probability that neither event occurs is 6 times the probability that both events occur.

The probability that neither event occurs = (1-A)(1-B)
The probability that both events occur = A*B
So (1-A)(1-B)=6AB......1-A-B+AB=6AB
Since we are looking at value of B, let us get the equation in terms of B lone.
1-B=5AB+A.......\(A=\frac{1-B}{5B+1}\)...(i)

Quote:
Furthermore, the probability that event B occurs and event A does not occur is 50 percent greater than the probability that event A occurs but event B does not occur.

The probability that event B occurs and event A does not occur = B*(1-A)
The probability that event A occurs but event B does not occur = A(1-B)
So, \(B(1-A)=\frac{3}{2}*A(1-B)\)..........
\(2B-2AB=3A-3AB.......2B=3A-AB\).......
A=\(\frac{2B}{3-B}\)...(ii)

Equating (i) and (ii)
\(A=\frac{1-B}{5B+1}=\frac{2B}{3-B}\).........
\((1-B)(3-B)=2B(5B+1)\).........
\(3-4B+B^2=10B^2+2B\).......
\(9B^2+6B-3=0.........3B^2+2B-1=0\)
\(3B^2+3B-B-1=0.......3B(B+1)-1(B+1)=0.......(3B-1)(B+1)=0\)

So B=-1 or B=\(\frac{1}{3}\)

As B cannot be less than 0, B=\(\frac{1}{3}\)

C


chetan2u - you're a god. Even after having a 720, I'm going for my 6th attempt at the GMAT and in one single question you cleared up my concepts of probability, set theory, and word problems all at once. Thank you
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Re: Events A and B occur independently of one another. [#permalink] New post   12 Jun 2022, 23:18
Expert Reply
twobagels wrote:
Events A and B occur independently of one another. The probability that neither event occurs is 6 times the probability that both events occur. Furthermore, the probability that event B occurs and event A does not occur is 50 percent greater than the probability that event A occurs but event B does not occur. What is the probability that event B occurs?

A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4


It's probably best to do the real math on this one, but just for anyone who gets stuck sometimes, we can also get there by Plugging In The Answers (PITA). I like trying B and D.

Let's start with B.
B = 1/4, so -B = 3/4
(1-A)(3/4)=6(A)(1/4)
(3-3A)/4=6A/4
3-3A=6A
3=9A
A=1/3, so -A = 2/3
What's the probability that B and -A? (1/4)(2/3) = 2/12 = 1/6
What's the probability that A and -B? (1/3)(3/4) = 1/4
Is 1/6 50% greater than 1/4? No, it's less than 1/4.
B is wrong.

Let's try D.
B = 1/2, so -B = 1/2
(1-A)(1/2)=6(A)(1/2)
(1-A)/2=6A/2
1-A=6A
1=7A
A=1/7, so -A = 6/7
What's the probability that B and -A? (1/2)(6/7) = 6/14
What's the probability that A and -B? (1/7)(1/2) = 1/14
Is 6/14 50% greater than 1/14? No, it's 500% greater.
D is wrong.

In B, the red was too small compared to the green.
In D, the red was too large compared to the green.
We need something in the middle.

Answer choice C.
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Re: Events A and B occur independently of one another. [#permalink] New post   13 Jun 2022, 06:53
twobagels wrote:
Events A and B occur independently of one another. The probability that neither event occurs is 6 times the probability that both events occur. Furthermore, the probability that event B occurs and event A does not occur is 50 percent greater than the probability that event A occurs but event B does not occur. What is the probability that event B occurs?

A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4


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Re: Events A and B occur independently of one another. [#permalink] New post   23 Apr 2024, 03:55
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