The events A and B are independent. The probability that both events A

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?

a) 0.58
b) 0.79
c) 0.84
d) 0.89
e) 0.93

a) 0.58
 b) 0.79
 c) 0.84
 d) 0.89
 e) 0.93

GmatDaddy · Accepted Answer

To Calculate: 
The probability that at least one of the events A and B occur.
Which will be equivalent to :
Probability that Event A occurs and B does not occurs + Probability that Event B occurs and A does not occurs + Probability that Event A and B both occur .
=A*(1-B) + B*(1-A) + A*B

The probability that event A occurs - A
The probability that event B occurs - B

Given:  
The probability that both events A and B occur= 0.21 ==A*B........................................(i)
The probability that event A occurs and event B does not occur = 0.49 ==A*(1-B)...........(ii)

A*(1-B)=0.49
A-AB=0.49
A-0.21=0.49 Using (i)
Therefore, A=0.70
And Hence, B=0.30

Now, Calculating the probability that at least one of the events A and B occur
A*(1-B) + B*(1-A) + A*B
0.49 + 0.30*0.30 + 0.70+0.30
0.49 + 0.09 + 0.21
0.79

Alternatively, We can also solve by calculating the Probability that none of the Events A or B Occur 
The probability that at least one of the events A and B occur = 1- Probability that none of the Events A or B Occur
1 - (1-A)*(1-B)
1 - (1-0.7)*(1-0.3)
1 - 0.3*0.7
1 - 0.21
0.79

GMATGuruNY · Answer

To organize the data, we can use the following Double-Matrix:
 https://thumb.ibb.co/imMEvT/matrix_A_and_B_probability_0.png

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 
Since (both A and B) = 0.21, and (A but not B) = 0.49, we get:
 https://thumb.ibb.co/cHvXFT/matrix_A_and_B_probability_1.png

(both A and B) is equal to (A)(B):
(A)(B) = 0.21
(A but not B) is equal to (A)(1-B):
(A)(1-B) = 0.49.
If we divide the first equation by the second, we get:
 \frac{B}{(1-B)} = \frac{0.21}{0.49} 
 \frac{B}{(1-B)} = \frac{21}{49} 
 \frac{B}{(1-B)} = \frac{3}{7} 
 7B = 3 - 3B

10B = 3

B = 0.3

If we insert (total B) = 0.3 into the matrix and calculate the remaining values, we get:
 https://thumb.ibb.co/gL8BaT/matrix_A_and_B_probability_final.png

Since (neither A nor B) = 0.21, (at least A or B) = 1 - 0.21 = 0.79.

B

PKN · Answer

Let's write all the given data in mathematical notation:-

P(A)P(B)=0.21----------(1)
P(A)(1-P(B))=0.49------(2)
Question stem:- 
P(at least one of A & B occur)=1-P(neither A nor B occurs)

from(2), we have P(A)-P(A)P(B)=0.49
 Or,P(A)-0.21=0.49
 Or, P(A)=0.7

So P(B)= \frac{0.21}{0.7} =0.3

P(neither A)=1-P(A)=1-0.7=0.3
P(neither B)=1-P(B)=1-0.3=0.7

So, P(at least one of A & B occur)=1-P(neither A nor B occur)=1-0.3*0.7=1-0.21=0.79

Ans. B

JaeTPU · Answer

We are given:
P(A and B) = .21
P(A and not B) = .49

Then, we know P(A) = P(A and B) + P(A and not B) = .7

because A and B are independent 
P(A and B) =P(A)*P(B)= .21
Plug in .7 into P(A) we get P(A)*P(B)= .7*P(B) = .21 and we know P(B) = .3
Then P(not A and B) = P(B) - P(A and B) = .3 - .21 = .09

Now, probability of at least one event occur is P(A and not B) + P(not A and B) + P(A and B) = .49 + .09 + .21 =.79

CrackverbalGMAT · Answer

Since the events A and B are independent, the probability of both A and B occurring is the product of their individual probabilities.
I.e. P(A and B) = P(A) * P(B).

The questions says that this probability is 0.21. Therefore,
P(A) * P(B) = 0.21.

The probability that A occurs and B does not occur is 0.49. This means that P(A) = 0.49 + 0.21 = 0.7. Hence, P(B) = 0.3 since P(A) * P(B) = 0.21.
We can conclude that the probability that B occurs and A does not occur = 0.3 – 0.21 = 0.09.

Probability that at least one of A and B occur = P(A) + P(B) – P(A and B), which, on substitution of values yields 0.79.

Drawing a Venn diagram can be another way of solving this. When we plug in the above values into a Venn diagram, it looks like the one below and from the Venn diagram, it’s clear that the area inside the circles represents the probability that at least one of A and B occur, which is 0.79.

Attachment:

26th Sept 2019 - Reply 3.JPG [ 22.69 KiB | Viewed 29052 times ]

What’s interesting to note is that Probability that both A and B will not occur is also 0.21 since it is equal to (1-0.7) * (1-0.3), which is reflected in the Venn also.

Hope this helps!

GMATinsight · Answer

There are 4 possible cases of outcomes and A and B events together

1) A happens and B does not happen
2) A happens and B happens as well
3) A does not happen and B happens 
 4) A and B both do not happen

What we need is the combined probability of first three cases

Probability of first 3 cases  = 1 -  4th Case  = 1- 0.21  = 0.79

Answer: Option B

fireflyejd1

gmexamtaker1 · Answer

Hello  ScottTargetTestPrep ,

Could you please share with us your approach on this probability problem?

ScottTargetTestPrep · Answer

Solution:
 
Notice that the events “both A and B” and “event A occurs and event B does not occur” together make up event A. In other words, if we take an outcome of event A, that outcome is in one and only one of the two events. Thus:

P(A) = P(both A and B) + P(event A occurs and event B does not occur) = 0.21 + 0.49 = 0.7

Since A and B are independent, P(A and B) = P(A) * P(B). Substituting P(A and B) = 0.21 and P(A) = 0.7 in this equality, we obtain:

0.21 = 0.7 * P(B)

P(B) = 0.21/0.7 = 0.3

Finally, “at least one of the events A and B occur” is the event “A or B,” which can be calculated using the formula P(A or B) = P(A) + P(B) - P(A and B). We get:

P(A and B) = 0.7 + 0.3 - 0.21 = 0.79

Answer: B

adipisci · Answer

Here, the probability theory looks wrong. It should not be P(A or B) = P(A) + P(B) + P(A and B).
It should be P(A or B) = P(A) + P(B) - P(A and B).

So, it doesn't sound right though it matches an answer.

Because A and B are independent events, P(A) . P(B) = 0.21.
So, P(B) = 0.21/0.49 = 0.428.
So, P(A or B) = 0.49 + 0.43 - 0.21 = 0.71.

But it does not match any answer choice given.

Where am I wrong?

Krunaal · Answer

P(A) is  not  0.49

P(A)*P(B not occurring) = 0.49

P(B not occurring) = 1 - P(B)

=> P(A)*  = 0.49

=> P(A) - P(A)*P(B) = 0.49

Now, since P(A)*P(B) = 0.21

=> P(A) = 0.49 + 0.21

=> P(A) = 0.7

=> P(B) = 0.3

To find probability at least 1 event occurs = 1 - probability both events don't occur = 1 - 0.3*0.7 = 1 - 0.21 = 0.79

The events A and B are independent. The probability that both events A

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The events A and B are independent. The probability that both events A

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards