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A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 73% (01:24) correct 27% (01:32) wrong based on 947 sessions

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A basket contains 5 apples, of which 1 is spoiled and the rest are good. If Henry is to select 2 apples from the basket simultaneously and at random, what is the possibility that the 2 apples selected will include the spoiled apple?

A. 1/5
B. 3/10
C. 2/5
D. 1/2
E. 3/5
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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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20
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A basket contains 5 apples of which one is rotten. If Henry is to select 2 apples from the basket simultaneously and at random what is the probability that the 2 apples selected will include the rotten apple?

A. 1/5
B. 3/10
C. 2/5
D. 1/2
E. 3/5

Direct combinatorial approach:

$$Probability=\frac{# \ of \ favorable \ outcomes}{total \ # \ of \ outcomes}$$:

Favorable outcomes: $$C^1_1*C^1_4$$ --> $$C^1_1$$ choosing one rotten out of 1 and $$C^1_4$$ choosing any for the second apple;

Total # of outcomes: $$C^2_5$$, choosing 2 apples out of 5;

$$P=\frac{C^1_1*C^1_4}{C^2_5}=\frac{2}{5}$$.

Reverse combinatorial approach:

Let's count the probability of the opposite event and subtract this value from 1. The opposit event will be if out of 2 apples both are good, so $$P=1-\frac{C^2_4}{C^2_5}=\frac{2}{5}$$.

Direct probability approach:

$$P=\frac{1}{5}*\frac{4}{4}+\frac{4}{5}*\frac{1}{4}=\frac{2}{5}$$: probability of {first apple rotten, second apple good} plus the probability of {first apple good, second apple rotten}.

Reverse probability approach:

Again 1- the probability of the opposite event --> $$P=1-\frac{4}{5}*\frac{3}{4}=\frac{2}{5}$$.

Check Probability chapter of Math Book for more on this issues: http://gmatclub.com/forum/math-probability-87244.html

Hope it helps.
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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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16
7
Total Apples = 5
Spoiled Apples = 1
Not Spoiled Apples = 4

Possible outcomes:
(Apple 1) Not Spoiled, (Apple 2) Spoiled; = (4/5)*(1/4) = 4/20 or 2/10.
OR
(Apple 1) Spoiled, (Apple 2) Not Spoiled; = (1/5)*(4/4) = 4/20 or 2/10.

Total Probability = 2/10 + 2/10 = 4/10 or 2/5.
##### General Discussion
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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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5
3
bmiller0731 wrote:
A basket contains 5 apples, of which 1 is spoiled and the rest are good. If Henry is to select 2 apples from the basket simultaneously and at random, what is the possibility that the 2 apples selected will include the spoiled apple?

a) 1/5
b) 3/10
c) 2/5
d) 1/2
e) 3/5

p (both apples are not spoiled) = 4C2/5C2 = 4*3/5*4 = 3/5

p (one of the apple spoiled) = 1-3/5 = 2/5
Intern  Joined: 18 Nov 2010
Posts: 8
Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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2
3
A basket contains 5 apples of which one is rotten. If Henry is to select 2 apples from the basket simultaneously and at random what is the probability that the 2 apples selected will include the rotten apple?

A. 1/5
B. 3/10
C. 2/5
D. 1/2
E. 3/5
Intern  Joined: 18 Nov 2010
Posts: 8
Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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This is the best explanation on probability I have ever got = using all the approaches. Thanks! I now know how Combinations/Direct Probability etc. all tie together!
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@ Bunuel, you're awesome. Do you focus on anything else apart form solving Quant questions on GC?:)
Just kidding. Kudos +1..
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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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Bunuel, you are awesome!
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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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+1 C

Total number of combinations: 5!/3! = 20
Number of combinations that include the spoiled apple:
When the spoiled apple is the first apple picked: 1*4= 4
When the spoiled apple is the second apple picked: 4*1 = 4
Total: 8

Probability: 8/20 = 2/5

C.
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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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bmiller0731 wrote:
A basket contains 5 apples, of which 1 is spoiled and the rest are good. If Henry is to select 2 apples from the basket simultaneously and at random, what is the possibility that the 2 apples selected will include the spoiled apple?

A. 1/5
B. 3/10
C. 2/5
D. 1/2
E. 3/5

Computing directly probabilities and using the complementary event:

P(spoiled apple selected) = 1 - P(no spoiled apple selected) = 1 - P(first apple not spoiled)*P(second apple also not spoiled) = 1 - (4/5)*(3/4) = 1 - 3/5 = 2/5.

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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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1
No. of apples that are not spoiled = 4
No. of apples that are spoiled = 1

The question is asking us to find out the probability of finding one spoiled apple among the 2 randomly selection ...

We can have 2 scenarios , Scenario A) where the first apple selected is not spoiled , and the second apple selected is spoiled and Scenario B) where the first apple selected is spoiled and the second selected is not spoiled ..

Scenario A) We have four apples out of five that are not spoiled , therefore probability is 4/5 , the second apple is a spoiled one out of the remaining 4 , therefore we have overall probability of this outcome as (4/5) x (1/4) = 4/20

Scenario B ) We have only one apple out of 5 that is spoiled so probability is 1/5 , the second apple can be one from any 4 remaining thefore 4/4... Probability of this outcome is (1/5) x (4/4) = 1/5 ....

We must add these two scenarios therefore we get 4/20 + 1/5 = 8/20 = 2/5 (C)
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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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Anagram helps.
Attachments Selecting 2 from 5 apples.jpg [ 26.71 KiB | Viewed 52762 times ]

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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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1
bmiller0731 wrote:
A basket contains 5 apples, of which 1 is spoiled and the rest are good. If Henry is to select 2 apples from the basket simultaneously and at random, what is the possibility that the 2 apples selected will include the spoiled apple?

A. 1/5
B. 3/10
C. 2/5
D. 1/2
E. 3/5

We need to determine the probability of selecting a spoiled apple and a non-spoiled apple when selecting two apples.

The number of ways to select the spoiled apple is 1C1 = 1. The number of ways to select a good apple is 4C1 = 4. Thus, the spoiled apple and a good apple can be selected in 1 x 4 = 4 ways.

The number of ways to select 2 apples from 5 is 5C2 = (5 x 4)/2! = 20/2 = 10.

Thus, the probability of selecting the spoiled apple and a good apple is 4/10 = 2/5.

Alternate Solution:

There are two outcomes that satisfy the requirement that the spoiled apple (S) is chosen along with a good apple (G), either S-G or G-S. The probability of S-G is (1/5)(4/4) = 1/5. The probability of G-S is (4/5)(1/4) = 1/5. Since either outcome satisfies our requirement, we add these two probabilities: 1/5 + 1/5 = 2/5.

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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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Good Apple= A
Spoiled Apple = S

Total ways of selecting two appls = ( A * S) + ( S*S)

What i want to ask is why are we not consideringt the order of selection here. For example ( A*S) could also be taken as ( S*A). Why is the order not improtant here?

Is it because the word [ Selection ] has been mentioned and that makes this a combination question?
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GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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Hi All

We're told that we have 4 regular apples and 1 spoiled apple. We're asked - if you grab 2 apples, then what's the probability of getting the spoiled apple?

The question can be solved in a couple of ways. Here's how to track each of the different outcomes that matches what we're looking for:

1st regular, 2nd spoiled = (4/5)(1/4) = 4/20
1st spoiled, 2nd regular = (1/5)(4/4) = 4/20

Total ways to get 1 spoiled and 1 regular = 4/20 + 4/20 = 8/20 = 2/5

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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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Top Contributor
bmiller0731 wrote:
A basket contains 5 apples, of which 1 is spoiled and the rest are good. If Henry is to select 2 apples from the basket simultaneously and at random, what is the possibility that the 2 apples selected will include the spoiled apple?

A. 1/5
B. 3/10
C. 2/5
D. 1/2
E. 3/5

The two approaches shown above are the best (faster) approaches, but here's one more.
Since there are so few objects involved (5 apples), we should be able to quickly answer the question by simply listing and counting

Let A, B, C, D, and E represent the 5 apples, and let E represent the SPOILED APPLE

We want to select 2 apples at random. So, let's list all of the possible outcomes:
1) AB
2) AC
4) AE
5) BC
6) BD
7) BE
8) CD
9) CE
10) DE

So, there are 10 possible outcomes
Of those 10 possible outcomes, 4 outcomes include the SPOILED APPLE

So, P(selection includes the spoiled apple) = 4/10 = 2/5

Cheers,
Brent
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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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bmiller0731 wrote:
A basket contains 5 apples, of which 1 is spoiled and the rest are good. If Henry is to select 2 apples from the basket simultaneously and at random, what is the possibility that the 2 apples selected will include the spoiled apple?

A. 1/5
B. 3/10
C. 2/5
D. 1/2
E. 3/5

$$5\,\,{\text{apples}}\,\,\left\{ \begin{gathered} \,1\,\,{\text{spoiled}} \hfill \\ \,4\,\,{\text{good}} \hfill \\ \end{gathered} \right.$$

$$? = P\left( {{\text{extract 2,}}\,\,{\text{1}}\,\,{\text{spoiled}}} \right)$$

$${\text{total}}:\,\,C\left( {5,2} \right) = 10\,\,\,{\text{equiprobables}}$$

$${\text{favorable:}}\,\,{\text{4}}\,\,\,\,\left( {{\text{spoiled}} + {\text{any}}\,\,{\text{good}}} \right)$$

$$? = \frac{4}{{10}} = \frac{2}{5}$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: A basket contains 5 apples, of which 1 is spoiled and the rest are  [#permalink]

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1
Bunuel wrote:
A basket contains 5 apples of which one is rotten. If Henry is to select 2 apples from the basket simultaneously and at random what is the probability that the 2 apples selected will include the rotten apple?

A. 1/5
B. 3/10
C. 2/5
D. 1/2
E. 3/5

Direct combinatorial approach:

$$Probability=\frac{# \ of \ favorable \ outcomes}{total \ # \ of \ outcomes}$$:

Favorable outcomes: $$C^1_1*C^1_4$$ --> $$C^1_1$$ choosing one rotten out of 1 and $$C^1_4$$ choosing any for the second apple;

Total # of outcomes: $$C^2_5$$, choosing 2 apples out of 5;

$$P=\frac{C^1_1*C^1_4}{C^2_5}=\frac{2}{5}$$.

Reverse combinatorial approach:

Let's count the probability of the opposite event and subtract this value from 1. The opposit event will be if out of 2 apples both are good, so $$P=1-\frac{C^2_4}{C^2_5}=\frac{2}{5}$$.

Direct probability approach:

$$P=\frac{1}{5}*\frac{4}{4}+\frac{4}{5}*\frac{1}{4}=\frac{2}{5}$$: probability of {first apple rotten, second apple good} plus the probability of {first apple good, second apple rotten}.

Reverse probability approach:

Again 1- the probability of the opposite event --> $$P=1-\frac{4}{5}*\frac{3}{4}=\frac{2}{5}$$.

Check Probability chapter of Math Book for more on this issues: http://gmatclub.com/forum/math-probability-87244.html

Hope it helps.

P=15∗44+45∗14=25P=15∗44+45∗14=25: probability of {first apple rotten, second apple good} plus the probability of {first apple good, second apple rotten}.

Hello Bunuel

Why are we considering the order of selection here....when do we know to consider order and when not too? Thanks Re: A basket contains 5 apples, of which 1 is spoiled and the rest are   [#permalink] 12 Oct 2019, 09:03
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