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# If four integers, a, b, c, and d are chosen at random with replacement

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Math Expert
Joined: 02 Sep 2009
Posts: 47183
If four integers, a, b, c, and d are chosen at random with replacement  [#permalink]

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11 Apr 2018, 21:31
10
00:00

Difficulty:

95% (hard)

Question Stats:

27% (01:17) correct 73% (01:27) wrong based on 79 sessions

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If four integers, a, b, c, and d are chosen at random, with replacement, what is the probability that ab + cd is even?

A. 1/4

B. 3/8

C. 1/2

D. 5/8

E. 3/4

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If four integers, a, b, c, and d are chosen at random with replacement  [#permalink]

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12 Apr 2018, 00:05
1
Bunuel wrote:
If four integers, a, b, c, and d are chosen at random, with replacement, what is the probability that ab + cd is even?

A. 1/4

B. 3/8

C. 1/2

D. 5/8

E. 3/4

We need to know the following rules in order to solve this problem:
Even*Odd = Odd*Even = Even*Even = Even | Odd*Odd = Odd
Even + Odd = Odd + Even = Odd | Odd + Odd = Even + Even = Even

---a-----b-----c-----d------ab+cd--
---E-----E-----E-----E---------E-----
---E-----E-----E-----O---------E-----
---E-----E-----O-----E---------E-----
---E-----E-----O-----O--------O-----
---E-----O-----E-----E---------E-----
---E-----O-----E-----O---------E-----
---E-----O-----O-----E---------E-----
---E-----O-----O-----O---------O-----
---O-----E-----E-----E----------E-----
---O-----E-----E-----O----------E-----
---O-----E-----O-----E----------E-----
---O-----E-----O-----O----------O-----
---O-----O-----E-----E----------O-----
---O-----O-----E-----O----------O-----
---O-----O-----O-----E----------O-----
---O-----O-----O-----O----------E-----

There are a total of 16 possibilities, of which the sum of ab and cd is even in 10 possibilities

Therefore, there are $$\frac{10}{16} = \frac{5}{8}$$ possibilities(Option D) when ab+cd is even
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Joined: 16 Mar 2018
Posts: 47
Location: United Kingdom
GMAT 1: 650 Q45 V35
Re: If four integers, a, b, c, and d are chosen at random with replacement  [#permalink]

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12 Apr 2018, 07:11
Oh wow I could not have thought of 16 possibilities like that...
My mind went like this
X=ab and Y=cd
For Sum of x and Y to be even it's possible in 2 cases
ODD +Odd is Even
Even + Even is Even
And other possibilities are
Even+Odd= Odd
Odd+ Even= Odd
So of Total 4 possibilities irrespective of what ab and cd could be the possibility fortheir summation to be even is 2/4 =1/2

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Joined: 24 Mar 2018
Posts: 64
If four integers, a, b, c, and d are chosen at random with replacement  [#permalink]

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25 Apr 2018, 07:41
pushpitkc Bunuel
What's wrong in my thinking ?
A number can be either odd or even so the probabiblity is half i.e. 1/2
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If four integers, a, b, c, and d are chosen at random with replacement  [#permalink]

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25 Apr 2018, 09:29
teaserbae wrote:
pushpitkc Bunuel
What's wrong in my thinking ?
A number can be either odd or even so the probabiblity is half i.e. 1/2

teaserbae

As you have clearly mentioned, the probabilty of a number being odd/even is $$\frac{1}{2}$$

Here we are concerned about the probabilities of 4 numbers.
Therefore, the overall probability will have $$2^4$$ or $$16$$ possibilities

Hope this helps you!
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If four integers, a, b, c, and d are chosen at random with replacement &nbs [#permalink] 25 Apr 2018, 09:29
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