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May 22nd, 2018, 09:00 AM   #1
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Probability 1

Hello

There is an exercise for which I'm not sure about my answers, so I'll be happy if you'll write please whether the answers are correct (and help/fix me please if the answers aren't correct).

In a certain town, the houses are built by three construction companies, which are A, B and C.
40% of the houses are built by A, 50% of these houses are for sale and the rest for rent.
30% of the houses are built by B, 60% of these houses are for sale and the rest for rent.
The rest of the houses are built by C, 50% of these houses are for rent.

a) What is the percentage of houses for sale of houses that are built by C?
b) A house for sale was selected. What is the probability that it was not built by A?
c) The events (probability theory) "The house was built by B" and "The house is for rent" are:
Independent events? Explain.
Foreign events? Explain.

My answers:
a) We know that 50% of houses of C are for rent, therefore, the houses of C for sale of all houses of C, is 50%.
b) A house for sale was selected, therefore, we will only relate to houses for sale. 20% of houses of A are for sale, 18% of houses of B are for sale and 15% of houses of C are for sale, so the answer is (18%+15%) / (20%+18%+15%), which is 33%/53% = 0.33/0.53 = 0.622
c) Yes, the events are independent events, because the company can build the houses which are for sale.
No, the events are not foreign, because the company can build the houses which are for rent.


If you didn't understand something, then tell me please and I'll explain.
And sorry for my bad English.

Thanks!

Last edited by greg1313; May 24th, 2018 at 06:26 AM.
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May 22nd, 2018, 09:41 AM   #2
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a) is not correct.

Suppose there are $N$ houses in total.

The total number of houses for sale built by $C$ is

${NS}_C = N(0.3)(0.5) = 0.15 N$

The total number of houses for sale is

$NS = N\left[(0.4)(0.5)+(0.3)(0.6)+(0.3)(0.5)\right] = 0.53N$

$P[\text{house for sale was built by C}] = \dfrac{{NS}_C}{NS}=\dfrac{0.15N}{0.53N} \approx 0.283$

and thus 28.3% of the houses for sale are built by C.

Last edited by romsek; May 22nd, 2018 at 09:53 AM.
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May 22nd, 2018, 09:47 AM   #3
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b) is correct
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May 22nd, 2018, 09:51 AM   #4
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for (c) you need to do better than this.

You need to determine whether or not

$P[\text{"house was built by B" AND "house is for rent"}]=P[\text{house was built by B}]P[\text{house is for rent}]$

I'm not familiar with (and neither does google seem to be) with the term "foreign event". Could you define this?
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May 22nd, 2018, 10:38 AM   #5
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Quote:
Originally Posted by romsek View Post
a) is not correct.

Suppose there are $N$ houses in total.

The total number of houses for sale built by $C$ is

${NS}_C = N(0.3)(0.5) = 0.15 N$

The total number of houses for sale is

$NS = N\left[(0.4)(0.5)+(0.3)(0.6)+(0.3)(0.5)\right] = 0.53N$

$P[\text{house for sale was built by C}] = \dfrac{{NS}_C}{NS}=\dfrac{0.15N}{0.53N} \approx 0.283$

and thus 28.3% of the houses for sale are built by C.
In a, I meant that the houses (of C) for sale, out of the all houses built only by C (sorry if I didn't explain it correctly).
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May 22nd, 2018, 10:43 AM   #6
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Quote:
Originally Posted by romsek View Post
for (c) you need to do better than this.

You need to determine whether or not

$P[\text{"house was built by B" AND "house is for rent"}]=P[\text{house was built by B}]P[\text{house is for rent}]$

I'm not familiar with (and neither does google seem to be) with the term "foreign event". Could you define this?
foreign events, strangers events.
The definition is that (for example) the two events can't happen together, in other words, if the event of A happened, then the event of B can't happen.
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May 22nd, 2018, 10:53 AM   #7
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Quote:
Originally Posted by IlanSherer View Post
foreign events, strangers events.
The definition is that (for example) the two events can't happen together, in other words, if the event of A happened, then the event of B can't happen.
ah... generally known as "mutually exclusive"

Clearly they aren't mutually exclusive as there is a 40% chance that any house B builds is for rent. (which you noted in your original post, well done.)
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May 22nd, 2018, 10:56 AM   #8
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Quote:
Originally Posted by IlanSherer View Post
In a, I meant that the houses (of C) for sale, out of the all houses built only by C (sorry if I didn't explain it correctly).
I see. You have the correct answer but the wrong reasoning.

You are directly told that the number of houses for sale is 50%.

From that you can derive that the number of houses for rent is 50%.

You did that backwards.
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May 24th, 2018, 03:48 AM   #9
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Imagine 1000 houses built.

"In a certain town, the houses are built by three construction companies, which are A, B and C.
40% of the houses are built by A, 50% of these houses are for sale and the rest for rent.

So 400 houses are built by A and 200 of those are for sale, 200 for rent.

30% of the houses are built by B, 60% of these houses are for sale and the rest for rent.
300 houses are built by B, 180 for sale and 120 for rent.

The rest of the houses are built by C, 50% of these houses are for rent.
300 of the houses are built by C, 150 for sale, 150 for rent.

"a) What is the percentage of houses for sale of houses that are built by C?"
There were a total of 200+ 180+ 150= 530 house for sale, 150 of them built by C. That is 150/530= 15/53= 28.3%.

"b) A house for sale was selected. What is the probability that it was not built by A?"
Again there were 530 houses for sale, 180+ 150= 330 not built by A. Given that a house is for sale, the probability it was not built by A is 330/530= 33/53= 62.2%.

"c) The events (probability theory) "The house was built by B" and "The house is for rent" are:
Independent events? explain."
Two events, P and Q, are "independent" if and only if the probability that P and Q both occur is equal to the probability P occurs times the probability Q occurs.
The probability the house was built by B is given as 30%. There were a total of 1000 houses built and 200+ 120+ 150= 470 are for rent. The probability that a house is for rent is 470/1000= 47/100= 47%. Of the 1000 houses built, 120 were built by B and were for rent. The probability "the house was built by B and it is for rent" is 120/1000= 12/100= 12%. 0.12 is NOT .3(.47)= 0.141 so these are NOT independent events.

"Foreign events? explain"
I presume that, by "Foreign events", you mean what I would call "mutually exclusive events"- they never occur together. Here the two events are "the house is built by B" and "the house is for rent". Since B did build some houses for rent, no, these are NOT "mutually exclusive events".

Last edited by Country Boy; May 24th, 2018 at 03:50 AM.
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