Discussion Problem_(27)

[duck]

Let, S be the sample space. Let, A and B be events. State true or false(provide a counter example if false).
a) If P(A)≤ P(B), then A⊆B
b) Assuming P(B)>0, P(A|B) is atleast as large as P(A).

[Mauli]

 first should be false
example: in three tosses of a coin:
pr(atleast two heads)
A={HHH,HHT,HTH,THH }= 4/8=1/2=PR(A)
PR(ATMOST TWO HEADS) =
HTH,THH,TTT,TTH,THT,HTT ,HHT}  PR(B)=7/8

PR(A)<PR(B)
BUT A is not a subset of B.
 
2) p(A/B) >=P(A)  this should also be FALSE
as P(A/B)<=P(B) IS TRUE.
PROOF:
n(A INTERSECTION B) <=n(A).. WHere n is the no of elements.
            n(B)             <=n(S)

DIVIDING THE ABOVE TWO INEQUALITIES WE GET
 P(A/B)<=P(A).
ALSO, a counter-example to prove 2nd is false.
let s be two tosses of a fair coin
A: head on the first toss
{hh,ht} pr(A)=2/4=1/2
B: tail
{tt,ht,th} PR(b)=3/4
pr(A intersection B) =1/4
PR(A/B) =1/4 / 3/4 = 1/3
PR(A) =1/2
THerefore , PR(A/B) is atleast as large as p(A)  
ie: p(A/B) >=P(A) IS FALSE.
 

[rohan]

SAME AS MAULI ..BOTH FALSE

[laracroft]

Both are false:

Explanation- assume set S={1,2,3,4,5,6,7,8,9,10}
A={1,2,3,4,5}
B={3,4,5,6,7,8,9,10}

(i) P(A)<=P(B) then A is subset of B
FALSE- here P(A)<=P(B) but A is not a subset of B

(ii) Assuming P(B)>0, P(A|B) is at least as large as P(A)
FALSE - here P(B)>0 i.e. 8/10, P(A|B) is 3/8 which is less than P(A).

Hence both false

[Sumit]

@Duck:..I'm also getting the both false..In fact it's quite easy to prove them false by taking random examples.... but could you Prove first part Mathematically...It would be great help....Thanks in advance..

[duck]

Correct Answers:
Both are false.

@ Sumit.. :)
First part cannot be proved as its false. You can only prove those things which are true.. And there are many random examples to support that its false.. :)