Prob & Stats! Old quiz questions...FINAL tomorrow. I don't understand how to do these?!?

Question

Value of x 2 3 4 5 6 7 8

Prob 1/16 2/16 3/16 4/16 3/16 2/16 1/16

x are outcomes of rolling a pair of four-sided dice.

Events s and t are independent with P(s) 1. 1/25
2. 6/25
3. 1/5
4. 2/5

Event a will occur with probability 0.5 and event b will occur with probability 0.6, P(a and b) = .0. the conditional probability of a, given b is
1. 0.3
2. 0.2
3. 1/6
4. Cannot be determined

Event a occurs with probability 0.3 and event b occurs with probability 0.4. if a and b are independent, we may conclude
1. P(a and b) = .12
2. P(a|b) = 0.3
3. P(b|a) = .4
4. All of the above

Answer 1

first let's talk about P(s|t),
this mean probability that the event s occurs knowing that the event t occured

If event s and t are independent, the probability that the event s occurs knowing that the event t occured is equal to probability that the event s occurs

ex. the probability that you get a A is maths knowing the you have a cat is equal to the probability that you get a A.

now if the 2 events are linked
probability that the event s occurs knowing that the event t occured= probability that the 2 events occurs knowing that the event t occued this mean P(s and t)/P(t)

ex. the probability that you get a A is maths knowing the you have listened to your teacher is equal to the probability that you get a A and you listened to your teacher divided by the probabilite you listened to you teacher




Events s and t are independent with P(s) 1. 1/25
2. 6/25
3. 1/5
4. 2/5

If s and t are independent,
P(s and t) = P(s)* P( t)
P(s|t)= P(s )
P(t|s)= P(t)

P(s)* P( t)=6/25
P(t)=6/25/p(s)

P(s|t)+P(t|s) =1= P(s)+ P( t)
P(t)= 1-P(s)
6/25/p(s)= 1-P(s)
6/25= P(s)-P(s)^2
P(s)^2-P(s)+6/25=0
P(s)=2/5 or 3/5
If p(s)=3/5 then p(t)=1-3/5=2/5
If p(s)=2/5 then p(t)=1-2/5=3/5
P(s)

Event a will occur with probability 0.5 and event b will occur with probability 0.6, P(a and b) = 0. the conditional probability of a, given b is
1. 0.3
2. 0.2
3. 1/6
4. Cannot be determined

That I don't get because P(a|b)= P(a and b)/ P(b)=0/.6=0

Event a occurs with probability 0.3 and event b occurs with probability 0.4. if a and b are independent, we may conclude
1. P(a and b) = .12
P(a and b) = P(a)* P( b)=.3*.4=.12
True

2. P(a|b) = 0.3
P(a|b)=p(a) = 0.3
True

3. P(b|a) = .4
P(b|a)=p(b) = 0.4
True

4. All of the above

Answer 2

Use the definition of conditional probability:
p(a|b)=p(a and b)/p(b)

first prob: p(s|t)=6/25 / p(t)
p(t|s)=6/25 / p(s)
so, adding them up...
(6/25) * (1/p(t) +1/p(s))=1
1/p(t)+1/(p(s)=25/6
1/(p(s)= 25/6 -1/p(t)
biggest p(t) can be is one, so
1/(p(s) <=19/6
p(s) >=6/19
and (4) =2/5 is the only given answer that meets this criterion.

Problem 2 must have a typo. If p(a and b ) were given, p(a|b) would be easy to compute using the definition of cond prob, above. If P(a and b ) is zero, the only answer would be zero.

Last question answer is (4). independent probabilities multiply when forming p(a and b ) = p(a) * p(b), and of course, independent means that conditional probabilities are just equal to the individual probabilities

Answer 3

These are all based on a couple of pretty simple concepts.

Two events s and t are independent if and only if P(s and t) = P(s)*P(t).

Conditional probability: P(s|t) = P(s and t)/P(t)

So, in particular, if s and t are independent,

P(s|t) = P(s and t)/P(t) = P(s)*P(t)/P(t) = P(s)

You should now be able to show that the answer to the first question is #4, 2/5.

You apparently have a typo in the next question. If P(a and b) = 0, a and b are mutually exclusive, and P(a|b) = 0. Given the correct value for P(a and b), the answer is
P(a and b)/0.6

Again, from the brief discussion above, you should be able to show that the answer is #4, all of the above.