Johnson has a corporate proposal. The probability that vice-president

Question

Johnson has a corporate proposal. The probability that vice-president Adams will approve the proposal is 0.7. The probability that vice-president Baker will approve the proposal is 0.5. The probability that vice-president Corfu will approve the proposal is 0.4. The approvals of the three VPs are entirely independent of one another.

Suppose Johnson must get the approval of at least two of the three VPs to win funding. What is the probability that Johnson’s proposal is funded?

(A) 0.14

(B) 0.26

(C) 0.49

(D) 0.55

(E) 0.86

Bunuel · Answer

Similar questions to practice: https://gmatclub.com/forum/johnson-has- ... 94287.html https://gmatclub.com/forum/johnson-has- ... 94288.html https://gmatclub.com/forum/johnson-has- ... 38926.html

stne · Answer

These are all the cases where Y voted and N did not vote for Mr. Adam, Mr. Baker and Mr. Corfu respectively: YYN+NYY+YNY+YYY

\frac{7}{10}*\frac{5}{10}*\frac{6}{10}+\frac{3}{10}*\frac{5}{10}*\frac{4}{10}+\frac{7}{10}*\frac{5}{10}*\frac{4}{10}+\frac{7}{10}*\frac{5}{10}*\frac{4}{10}

.21+.06+.14+.14

=.55

Ans- D

Hope it's clear.

Fdambro294 · Answer

Probability that at least 2 of 3 approve =

(1) — (Probability of either: only 1 approves OR none approve)

P(A) = .7
P(B) = .5
P(C) = .4

Case 1: probability only 1 approves

(.7) (.5) (.6) 
+
(.3) (.5) (.6)
+
(.3) (.5) (.4) =

(.21) + (.09) + (.06) =

.36

Case 2: probability NONE approve

(.3) (.5) (.6) =

.09

Probability of Case 1 OR Case 2 = (.36) + (.09) = .45

1 - (.45) = .55

*answer*
.55
D

Posted from my mobile device

Fdambro294 · Answer

Probability that at least 2 of 3 approve =

(1) — (Probability of either: only 1 approves OR none approve)

P(A) = .7
P(B) = .5
P(C) = .4

Case 1: probability only 1 approves

(.7) (.5) (.6) 
+
(.3) (.5) (.6)
+
(.3) (.5) (.4) =

(.21) + (.09) + (.06) =

.36

Case 2: probability NONE approve

(.3) (.5) (.6) =

.09

Probability of Case 1 OR Case 2 = (.36) + (.09) = .45

1 - (.45) = .55

*answer*
.55
D

Posted from my mobile device

KevoK · Answer

I wonder if there's a more effective way to solve this other than just doing it the calculation heavy way.

vallequalle · Answer

I dont think you can do it without those heavy calculations but since there is a 0.5 in every calculation you could save some time with neglecting the 0.5 in the first place and just coming back to it in the end so you can come up to the answer in under a minute:

2 approve:

7*4 = 28
7*6 = 42
3*4 = 12

3 approve:

7*4 = 28

All added up = 110

Times 0.5 = 55

/100 = 0.55

egmat · Answer

Hi KevoK,

Great question — there IS a slightly slicker way to organize this. Instead of listing all  4  winning scenarios separately, try conditioning on one VP.

Pick the easiest VP to work with. Here, Baker is nice because P(B) =  0.5  and P(B') =  0.5 .

Split into two worlds:

World 1: Baker approves (probability 0.5) 
Johnson now only needs  1  more approval from Adams or Corfu. It's easier to find the  complement  — neither approves:  0.3  ×  0.6  =  0.18 . So P(at least  1  of them approves) =  1  -  0.18  =  0.82 .

World 2: Baker doesn't approve (probability 0.5) 
Johnson now needs BOTH Adams AND Corfu:  0.7  ×  0.4  =  0.28 .

Combine: 
 0.5  ×  0.82  +  0.5  ×  0.28  =  0.41  +  0.14  =  0.55

Answer: D

Why this is faster:   instead of computing  4  separate three-way products and adding them up, you only need  2  quick sub-calculations. The trick is to  condition on one person , which reduces the 'at least  2  out of  3 ' problem into simpler 'at least  1  out of  2 ' and 'both out of  2 ' problems.

General principle:   When a probability problem feels calculation-heavy, try splitting on one variable to break it into simpler sub-problems. This  'divide and conquer' approach  often cuts the work roughly in half.

Johnson has a corporate proposal. The probability that vice-president

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

Johnson has a corporate proposal. The probability that vice-president

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