Could I get help to solve this problem please
400 people were checked on about their involvement with neighbor hood
watch according to their street numbers.  It was also checked how many
neighbors had security alarm and how many don’t.  Results are as
follows (relevant frequencies).

Street number			Houses with security 		Houses without 
				Alarm				Alarm
Less than 10			0.10				0.65
10 or more			0.20				0.05

What would be the probability that the houses were without alarm?
Explain whether the events houses without alarm and street number for
less than 10 days are mutually exclusive
Determine if involvement with neighbor hood watch according to their
street numbers for 10 or more days independent of whether a house had
security alarm or did not have security alarm.

Please explain in detail how to do this 
Elshana

Hi elshana!
Here are the answers to your questions:


What would be the probability that the houses were without alarm?

We can think of each "square" as an event that can't happen at the
same time as any other. For example, the event "House w/alarm and less
than 10 days" can't of course happen for a house at the same time as
"House w/alarm and 10 or more" or at the same time as "House w/o alarm
and less than 10 days"; because it obviously either has or doesn't
have security, and it has been involved with the neighborhood watch
for either less than 10 or more than 10 days. This alse means that
these events are mutually exclusive (we'll see more about this in the
following question). It's important to remember that here we're
defining the "events" to be both the "alarm status" and "neighborhood
watch status". Since no two of these events can happen at the same
time, we can apply the following probability rule to answer the first
question:

"If you have two outcomes that can't happen at the same time, then the
probability that either outcome occurs is the sum of the probabilities
of the individual outcomes."

Probability Rules
http://www-math.bgsu.edu/~albert/m115/probability/prob_rules.html

Notice that the event "house without alarm" is equivalent to the
event: "(house w/o alarm and less than 10 days) OR (house w/o alarm
and 10 days or more)". Since these two events can't happen at the same
time, we can sum their probabilities to find the probability that
either happens; and this will be the probability that the house is
without alarm. Therefore,

Prob (house w/o alarm and less than 10 days) = 0.65
Prob (house w/o alarm and 10 days or more) = 0.05

==> Prob (house w/o alarm) = 0.65 + 0.05 = 0.70

So the answer is 0.70


Explain whether the events houses without alarm and street number for
less than 10 days are mutually exclusive.

"Two events are mutually exclusive if it is not possible for both of
them to occur"

Mutually exclusive events
http://www.ruf.rice.edu/~lane/hyperstat/A132677.html

Is this the case of the events "house w/o alarm" and "less than 10
days"? Clearly not. If they were mutually exclusive, then the
probability of both happening would be 0 (because the event couldn't
happen). However, we can see from the table that the probability of
both happening is 0.65. Therefore, they are not mutually exclusive.

Examples of mutually exclusive events would be "house w/ alarm" and
"house w/o alarm": these two can't happen at the same time for a
house. Another example could be "house w/alarm and less than 10 days"
and "house w/o alarm and 10 days or more". If a house has an alarm and
has been with the neighborhood watch for less than 10 days, it's
impossible for it to not have an alarm and be in the neighborhood
watch for more than 10 days.


Determine if involvement with neighbor hood watch according to their
street numbers for 10 or more days independent of whether a house had
security alarm or did not have security alarm.

As you can see in the following page, the definition of independent
events is:
"A and B are independent if 

Prob(A and B) = Prob(A)*Prob(B).

Independent Events
http://regentsprep.org/Regents/math/mutual/Lindep.htm

Let event A be "10 or more days" and event B be "house w/ alarm". We
know from the table that Prob(A and B) is 0.2. We must now find wether
this is equal to Prob(A)*Prob(B).

So what's Prob(A), or Prob(10 or more days)? Using the same reasoning
as in the first question, we find that this probability is
0.20+0.05=0.25. Similarly, Prob(B) or Prob(house w/alarm) is
0.10+0.20=0.30. Therefore, Prob(A)*Prob(B)=0.30*0.25=0.075. Clearly,
this is not equal to 0.2. Thus the events A and B (the events "10 or
more days" and "house w/ alarm") are not independent. In a similar
fashion it can be shown that the events "10 or more days" and "house
w/o alarm" are also non-independent.


Google search terms
"independent events"
"mutually exclusive"
probability introduction


I hope this helps! If you feel there's anything unclear in my answer,
please request a clarification before rating it. Otherwise, I await
your rating and final comments.

Best luck!
elmarto

your answer has certainly provided me a very clear picture of how to
tackle such problems
ElShana