Kalpan probability question

[priyasaibaba]

Can anyone help me solve this question?

The events A and B are independent. The probability that both events A and B occur is 0.21. The probability that event A occurs and event B does not occur is 0.49. What is the probability that at least one of the events A and B occur?

Can you also explain me the following? This statement was given in Kaplan explanation.

Now when the event A occurs, the event B either occurs or does not occur. So the probability that event A occurs is equal to the probability that both events A and B occur, plus the probability that event A occurs and event B does not occur.

I thought the probability for event A to occur is, Probability that event A only occurs + Probability that both events occurs.

Can someone clarify the above explanation given by Kaplan.

Thanks,
Priya

[amising6]

Can anyone help me solve this question?

The events A and B are independent.
( And B are not depending on each other)The probability that both events A and B occur is 0.21)p(a+B). The probability that event A occurs and event B does not occur is 0.49. What is the probability that at least one of the events A and B occur?


Can you also explain me the following? This statement was given in Kaplan explanation.

Now when the event A occurs, the event B either occurs or does not occur (reason being A is independent of B ,so if A occurs B can occur or cannot occcur). So the probability that event A occurs is equal to the probability that both events A and B occur, plus the probability that event A occurs and event B does not occur.( here they are taking about at least one of the events A and B occur)


I thought the probability for event A to occur is, Probability that event A only occurs ( you can not say only A as it is independent of B)+ Probability that both events occurs.


Can someone clarify the above explanation given by Kaplan.

Thanks,
Priya

yes

[KapTeacherEli]

Hi Priya,

The general principle of probabilities is that mutually exclusive probabilities add together. In other words, if I am picking a card out of a deck, the probability of getting a King is 4/52 = 1/13 (4 kings our of 52 cards). The probability of picking a Queen is the same. Thus, the probability of a single pick being a King or a Queen is 1/13 + 1/13 = 2/13; I cannot pick a King and a Queen at the same time, so I add the mutually exclusive probabilities. This matches what I would get using the standard probability formula; there are a total of 8 Kings and Queens in the deck, and 8/52 = 2/13.

Here, if event A happens, there are two mutually exclusive possibilities: either B happens or it doesn't. Thus, the odds of A happening and B happening and the odds of A happening and B not happening can be added; the result, 0.6, is the odds of A occurring and B either occurring or not.

Finally, we apply a little common sense; B occurring or not occurring are the only two possibilities, since the default assumption is that events can half-occur. Thus, since we've added the chances of A happening with B and A happening without B, we have accounted for every possible occurrence of A. That leaves us with 0.6 as the final value for the odds of A happening.

[ajayiitr]

The events A and B are independent. The probability that both events A and B occur is 0.21. The probability that event A occurs and event B does not occur is 0.49. What is the probability that at least one of the events A and B occur?

A and B are independent = probability of A happening is not affected by B happening, similarly probability of B happening is not affected by A happening. If on other hand, if probability of A or B is dependent on other, that will be conditional probability. May be GMAT does not test that.

However, they can occur together or not. That will be determined by whether they are mutually exclusive or not. If mutually exclusive, P(A and B) = 0, which is not situation here, as,

P(A and B) = 0.21 = P(A) X P(B)
The probability that event A occurs and event B does not occur is 0.49 implies: P(A=true and B = false) = P(A) X(1-P(B)) = 0.49

Question is: What is the probability that at least one of the events A and B occur? implies: P(A or B) = P(A) + P(B)- P(A and B) = P(A) + P(B) - P(A) X P(B)

Solving from given two eqns, my answer is 0.79

[gmat1011]

This formula triggers some memories (I remember this from math class in school!):

P(A or B) = P(A) + P(B)- P(A and B) = P(A) + P(B) - P(A) X P(B)

Can someone clarify if P(A or B) is the same as saying: "probability that at least one of the events A and B occur"

The problem for applying this formula is figuring out the p(B) --- which "ajayiitr" derived above based on p(A) which is given

You get the same result of .79 if you draw a venn diagram - put .21 in the overlapping region and put .49 (for the remaining part of circle A); the remaining part of circle B other than overlapping region is .3 (as .3 + .49 + .21 make 1).

Add .49 + .3 + = .79

Can anyone provide the correct answer - the Venn diagram method seems a bit simplistic and I am not sure if its correctly applied here...

[dku123]

Can anyone help me solve this question?

The events A and B are independent.
( And B are not depending on each other)The probability that both events A and B occur is 0.21)p(a+B). The probability that event A occurs and event B does not occur is 0.49. What is the probability that at least one of the events A and B occur?


Can you also explain me the following? This statement was given in Kaplan explanation.

Now when the event A occurs, the event B either occurs or does not occur (reason being A is independent of B ,so if A occurs B can occur or cannot occcur). So the probability that event A occurs is equal to the probability that both events A and B occur, plus the probability that event A occurs and event B does not occur.( here they are taking about at least one of the events A and B occur)


I thought the probability for event A to occur is, Probability that event A only occurs ( you can not say only A as it is independent of B)+ Probability that both events occurs.


Can someone clarify the above explanation given by Kaplan.

Thanks,
Priya

yes

This problem can also be solved by using 2*2 matrix(table)
A notA Total

B .21 .09 .30

notB .49 .21 .70

Total .70 .30 1

The probability that at least one event occurs= 1-the probability that neither of them occurs=1-.21=.79 OR
P(A OR B)=P(A)+P(B)-P(AB)=.70+.30-.21=.79,
I think this is helpful and much easier.
Kind Regards
Dinesh

[mparakala]

Lets denote A and B as follows:

probability of "A" OCCURRING = a
probability of "A" NOT OCCURRING = (1-a)

similarly, B occurring = b
B NOT occurring = (1-b)

given, ab = 0.21;

a * (1-b) = 0.49
=> a-ab = 0.49 (substitute ab=0.21)

a = 0.70 ; (1-a) = 0.30
b = 0.30 ; (1-b) = 0.70

ATLEAST one of A and B occurring = 1 - (Neither A nor B occurring)
= 1 - ([1-a] * [1-b])
= 1- (0.3*0.7)
= 1-0.21
Answer = 0.79

[Scott@TargetTestPrep]

Can anyone help me solve this question?

The events A and B are independent. The probability that both events A and B occur is 0.21. The probability that event A occurs and event B does not occur is 0.49. What is the probability that at least one of the events A and B occur?

Can you also explain me the following? This statement was given in Kaplan explanation.

Now when the event A occurs, the event B either occurs or does not occur. So the probability that event A occurs is equal to the probability that both events A and B occur, plus the probability that event A occurs and event B does not occur.

I thought the probability for event A to occur is, Probability that event A only occurs + Probability that both events occurs.

Can someone clarify the above explanation given by Kaplan.

Thanks,
Priya

Solution:

The probability that at least one of the events A and B occur is the same as saying the probability that A or B occurs. Similarly, the probability that event A occurs and event B does not occur is the same as saying the probability only A occurs. Recall that:

P(A or B) = P(A) + P(B) - P(A and B)

and

P(A only) = P(A) - P(A and B)

and if A and B are independent, as in this case, we have:

P(A and B) = P(A) x P(B)

From the second equation above, we have:

0.49 = P(A) - 0.21

0.7 = P(A)

From the third equation, we have:

0.21 = 0.7 x P(B)

0.3 = P(B)

Finally, using the first equation, we have:

P(A or B) = 0.7 + 0.3 - 0.21

P(A or B) = 0.79

Answer: B