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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
A. 1/5
B. 1/4
C. 1/3
D. 1/2
E. 3/4
22 Apr 2021, 00:47
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twobagels
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
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=\(\frac{1}{3}\)
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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
Posted from my mobile device
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
Posted from my mobile device
