Probability Statistics Question

[geforce]

In 2007, 52% of all immigrants to Canada were females, 25% were under 18
years old, and 12% were females under 18 years old.

a. Find the probability that a randomly selected person who immigrated to Canada in
2007 was a female and over 18 years old.

b. Find the probability that a randomly selected person who immigrated to Canada in
2007 was either a female or under 18 years old.

c. Find the probability that a randomly selected person who immigrated to Canada in
2007 was a male and over 18 years old.

d. Are sex and age independent events? Why? (Need a statistical reasoning.)

e. Are sex and age disjoint events? Why? (Need a statistical reasoning.)

I just don't understand the nature of this question, if someone could explain It would be very helpful. So far, I think this is what the questions should look like.
A) 52 + 12 = 64% or (.52 x .12 ) x 100 = 6.24%
B) 52% were females, 25% were under 18 years old so, (.52 x .25) x 100 = 13%
C) 100 - 52 = 48 % was male and the over 18 years old is where i got confused because of the nature of the question.
E) Tryed, but didn't understand.

 

[Ninty64]

Did you try drawing a Venn Diagram? The problem gives you information about two groups of people and their intersection. This is a good starting point.

 

[geforce]

I tryed that before and it didn't help much. I don't think I even understood it correctly.

 

[geforce]

anyone here?

 

[Ninty64]

A Venn Diagram for two groups of people would generally look like this:

The square would encompass the entire set, which in this case would be the immigrants to Canada in 2007. We know the following:
52% were females
25% were under 18 years old
12% were females under 18 years old. (intersection)

 

[geforce]

A) So,
52% are females
100 - 52 = 48 % are males
48 - 25 - 12 = 11% are females and are over 18 years old.

B) So for females its 52% and under 18 years old its 25% so it would be 77% that are females or under 18 years old.

C) 100 - 52 = 48% male
52 - 25 - 12 = 15% are males who are over 18 years old.

How would I find the statistical reason for D) and E) I know there's some math I have to do.

 

[geforce]

For D) I would use P(A l B ) = P(A) to prove if A and B are independent or the venn diagram and for E) I would prove if "S intersect A = 0 " or by ven diagram, two separate circles.

 

[geforce]

So for D) P(S l A) = P(`S) where `S means not S, to prove they are not indepedent. Since it's not easy to do through formula's the venn diagram helps by showing the intersection of the Sex(52%) and Age(Under 18 years old) So therefore, S intersect A is true so it's indepdent.

And for E) Sex and age are only disjoint events if and only if "S Intersect A = 0" Since you can't subtract them mathamatically this wouldn't make sense. Therefore, they are not disjoint events.

 

[geforce]

Although I think I'm done I still think B) is wrong, anyone have an idea?

 

[Ninty64]

A) So,
52% are females
100 - 52 = 48 % are males
48 - 25 - 12 = 11% are females and are over 18 years old.

The 12% of females under 18 is included in the 52% females. If you imagine the Venn Diagram, the female circle (x+y) must equal 52. Since once group of the females is under 18 (y), the other group must be over 18 (x).

B) So for females its 52% and under 18 years old its 25% so it would be 77% that are females or under 18 years old.

Very close. You have the right idea, you just added the same people (the intersection) twice. 52% are females and 25% are under 18. However, of the 52% females, 12% of them are under 18. Thus that 12% of under 18 females is included in both the 52% and 25%. By simply adding 52% and 25%, you added that 12% twice.

C) 100 - 52 = 48% male
52 - 25 - 12 = 15% are males who are over 18 years old.

48% of the population are males. That part is true. That 48% can be split into two groups: the males that are under 18 and the males that are over 18. We know that of the 25% of people under 18, 12% of them are females. Using that, we should be able to figure out how many are males, and consequenty, how many males are over 18.

How would I find the statistical reason for D) and E) I know there's some math I have to do.

I'm not positive about the last two, but I would approach them by definition. From what I know, two events are independent if the outcome of one event does not affect the outcome of another event. This means that the probability of both outcomes happening is the probability of one outcome times the probability of the other.

For the disjoint problem, I'm not sure. I would base my answer off Wiki's definition. Maybe someone else can provide more insight.

 

[geforce]

A) Since 52 - 12 = 40% which is 40% females that are over 18 years old.
B) If 52 - 12 = 40% that means 40% females are over 18 years old. Then, 12% are females under 18 years old.
C) Since we know 25 - 12 = 13% are females under 18 years old then 12% are males under 18 years old. So, 48 - 12 = 36%. Therefore, 36% are males over 18 years old.

Should be correct now

 

[geforce]

I'm not positive about the last two, but I would approach them by definition. From what I know, two events are independent if the outcome of one event does not affect the outcome of another event. This means that the probability of both outcomes happening is the probability of one outcome times the probability of the other.

For the disjoint problem, I'm not sure. I would base my answer off Wiki's definition. Maybe someone else can provide more insight.

I could use formulas as stated above, but how would I differentiate the Sex and Age as integers? The best way would be using venn diagrams for this kind of situation.

 

[Ninty64]

B) If 52 - 12 = 40% that means 40% females are over 18 years old. Then, 12% are females under 18 years old.

Yes, 40% are females over 18, 12% are females under 18, and how many are males under 18? Female OR under 18 includes all three categories

C) Since we know 25 - 12 = 13% are females under 18 years old then 12% are males under 18 years old. So, 48 - 12 = 36%. Therefore, 36% are males over 18 years old.

I made a typo. I said "male" where I meant "female" it should be 35% because 12% are females and 13% are males, not the other way around.

The Venn Diagram should have been as follows:

If you notice, once you make the Venn Diagram, it's much simpler to isolate groups and find what you're looking for.
The female circle should total 52 (40 + 12), the under 18 circle should total 25 (13 + 12), and the entire set should total 100 (40+12+13+35).

 

[Ninty64]

Quite frankly, I don't know how to answer E.

For D:
[/PLAIN]
When two events, A and B, are independent, the probability of both occurring is:
P(A and B) = P(A) · P(B)

I might try something along those lines.

Edit: But this is what I would do. I cannot assert the correctness of this!

 

[geforce]

Hmm , I don't understand where did you make the typo that says 12% are females because 25% - 12% = 13%

 

[Ninty64]

Hmm , I don't understand where did you make the typo that says 12% are females because 25% - 12% = 13%

I fixed it. In the original problem we were told that 12% were females under 18. That's was given. so 25% (of under 18) - 12% (of females under 18) = 13% (of males under 18)

 

[geforce]

Oh right haha.

So then

B) Okay, since 25 - 12 = 13% males under 18 years old
48 - 13 = 35%
Males OVER 18 : 35%
52 - 12 = 40
Females OVER 18: 40%

So finally, 100 - 40 - 35 = 25%
So, 25% are females and under 18 years old.
C) Oh okay I see that still makes a lot of sense, 25 - 12 = 13% males under 18 years old and 48 - 13 = 35% males over 18 years old.

 

[Ninty64]

B) Okay, since 25 - 12 = 13% males under 18 years old
48 - 13 = 35%
Males OVER 18 : 35%
52 - 12 = 40
Females OVER 18: 40%

So finally, 100 - 40 - 35 = 25%
So, 25% are females and under 18 years old.

What you just solved for was the amount of people under 18 there were.
100 - 40 (males > 18) - 35 (females > 18) = 25 (people < 18)
You solved for the wrong thing. The math is right though, since we were told that there are 25 people under 18, which is what you got. However, you should be solving for how many people are either female or under 18. Or means Union. Everyone who's female Union everyone who's under 18. Just make sure you don't count the females who are under 18 twice, like you did in your earlier attempt

I really think the Venn Diagram would help a lot here. It's good at visually presenting all of the information.

 

[geforce]

B)
What about:
25 - 12 = 13% Males under 18 years old
48 - 13 = 35% males OVER 18 years old

52 - 12 = 40% females under 18 years old
52 - 40 = 12% females OVER 18 years old
SO,
(52 - 40) + (48 - 35) = 25% females and under 18 years old.

Thanks a lot for your help i REALLY appreciated it thank you VERY much.

 

[Ray Vickson]

In 2007, 52% of all immigrants to Canada were females, 25% were under 18
years old, and 12% were females under 18 years old.

a. Find the probability that a randomly selected person who immigrated to Canada in
2007 was a female and over 18 years old.

b. Find the probability that a randomly selected person who immigrated to Canada in
2007 was either a female or under 18 years old.

c. Find the probability that a randomly selected person who immigrated to Canada in
2007 was a male and over 18 years old.

d. Are sex and age independent events? Why? (Need a statistical reasoning.)

e. Are sex and age disjoint events? Why? (Need a statistical reasoning.)

I just don't understand the nature of this question, if someone could explain It would be very helpful. So far, I think this is what the questions should look like.
A) 52 + 12 = 64% or (.52 x .12 ) x 100 = 6.24%
B) 52% were females, 25% were under 18 years old so, (.52 x .25) x 100 = 13%
C) 100 - 52 = 48 % was male and the over 18 years old is where i got confused because of the nature of the question.
E) Tryed, but didn't understand.

You can use a Venn diagram, as some have suggested, or you can use a tabular layout to help you get at the issues. We can think of the date about percentages as being laid out in a 2x2 table:

Male Female total
Younger x 12 25
older y z ?
total ? 52 100

So, x + 12 = 25 and 12 + z = 52. Can you see how to get the other two missing row and column totals? Can you see how to get the entries x, y and z? Knowing these will answer questions (a)--(c).

RGV