Thanks for the answer, if one does not know the number of houses in the suburb?
I only have one problem with answer b.
I think the events are probabilistically independent and cannot be settled with mathematical arguments and can only be settled with impirical data to decide wheater independence is reasonalbe.
They most certainly are
not independent. In order for
neither house to have a security system, two things must happen, the second being conditional on the first: the first house you pick must not have a security system (which obviously has a probability of 70%), AND, having
already picked a house that doesn't have security system, you must pick another house that also doesn't have a security system (that much is obvious, but what is not obvious is that the probability of this is
less than 70%).
A different scenario will make the reason clear: suppose you have two shoes, 50% of which are left shoes and 50% of which are right shoes. If you pick a shoe at random, what is the probability of getting a left shoe? 50%, right? Now, if you pick two shoes, what is the probability that
both of them are left shoes? I'll give you a hint: it's less than 25%.
To go back to the original question, after you've picked the first house, the second time you pick a house,
there is one less house to pick from (you can't pick the same house again because the question specifically requires you to pick
two houses, not one house twice). And, because the question requires the first house you picked to
not have a security system, of the houses that remain, you still have the same number that have security systems. For example, if there are 100 houses, 30 of which have a security system, then after picking a house that
doesn't have a security system, there are now only 99 houses left to pick from, but still 30 which have a security system (and therefore 69 that don't). So the probability of the second house
not having a security system is 69/99, or 69.697%. And so the total probability of both the first house not having a security system (70%), AND the second house also not having a security system (69.697%) is 70% x 69.697% = 48.788% The fewer houses there are, the lower the probability, with the probability reaching zero when there only two to chose from in the first place (as in the shoe example above).
