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Claire estimates her chances of being admitted to her top five schools as follows: 80 percent for Debs U, 70 percent for Powderly State, 50 percent for Randolph A&M, 20 percent for Reuther, and 10 percent for Chavez Poly. If Claire’s estimates are correct, her chance of being admitted to at least one of her top five schools is between

Claire estimates her chances of being admitted to her top five schools as follows: 80 percent for Debs U, 70 percent for Powderly State, 50 percent for Randolph A&M, 20 percent for Reuther, and 10 percent for Chavez Poly. If Claire’s estimates are correct, her chance of being admitted to at least one of her top five schools is between

A) 75% and 79.99%

B) 80% and 84.99%

C) 85% and 89.99%

D) 90% and 94.99%

E) 95% and 99.99%

When it comes to probability questions involving "at least," it's best to try using the complement. That is, P(Event A happening) = 1 - P(Event A not happening) So, here we get: P(admission to at least 1 school) = 1 - P(not getting admitted to at least 1 school) What does it mean to not get admitted to at least 1 school? It means getting admitted to zero schools. So, we can write: P(admission to at least 1 school) = 1 - P(getting admitted to zero schools)

P(getting admitted to zero schools) = P(not getting admitted to Debs U AND not getting admitted to Powderly State AND not getting admitted to Randolph A&M AND not getting admitted to Reuther AND not getting admitted to Chavez Poly) = P(not getting admitted to Debs U) x P(not getting admitted to Powderly State) x P(not getting admitted to Randolph A&M) x P(not getting admitted to Reuther) x P(not getting admitted to Chavez Poly) = 0.2 x 0.3 x 0.5 x 0.8 x 0.9

ASIDE: We can avoid tedious calculations by noticing that 0.2 x 0.3 x 0.5 ALREADY equals 0.03 So, further multiplying by 0.8 and 0.9 will make the product even smaller than 0.03 So, P(getting admitted to zero schools) = some number smaller than 0.03

P(admission to at least 1 school) = 1 - some number smaller than 0.03 = some number GREATER than 0.97

Answer: E

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Re: Claire estimates her chances of being admitted to her top f [#permalink]
26 Jul 2017, 06:08

4

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Expert's post

Claire estimates her chances of being admitted to her top five schools:

80 percent for Debs U, or 0.8

70 percent for Powderly State, or 0.7

50 percent for Randolph A&M, or 0.5

20 percent for Reuther, and or 0.2

10 percent for Chavez Poly. or 0.1

What are the chances for her getting invited to none of the colleges! \(P = (1- 0.8)\times(1- 0.7)\times(1- 0.5)\times(1- 0.2)\times(1- 0.1)= 0.2 \times 0.3 \times 0.5 \times 0.8 \times 0.9 = 0.0216\).

So her chance of sucsess at atleat one school is \((1-P)=0.9784\) or 97.84%

Hence option E is correct!
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