Question

A box has two balls, one white and one red. We select one ball, put it back in the box, and select a second ball (sampling with replacement). Let T be the event of getting the white ball twice, F the event of picking the white ball first, S the event of picking the white ball in the second drawing.

a. Compute P(T).

b. Compute P(T|F).

c. Are T and F independent?.

d. Are F and S mutually exclusive?

e. Are F and S independent?

Short answer

P(T) = $1/4$; P (T | F) =$1/4$

T & F are not independent ; F & S are not mutually exclusive ; F & S are independent.

Step-by-step solution

Probabilities calculation 

• $Pr\left(T\right)$= P (getting white ball twice)

Pr (getting white ball once) = $1/2$; P (get white ball again, with replacement) = $1/2$

So, $P\left(T\right)$= $1/2x1/2=1/4$

• $Pr(TIF)$= Pr getting white ball twice, given first ball is white

= $P(T\cap F)/P\left(F\right)$

= ( 1 /4 ) / ( 1 / 2 )

= 1 / 2

Independent & mutually exclusive events 

Independent Events are those where occurrence of one event doesn't effect the occurrence of other event. Mutually Exclusive events are those which cannot happen simultaneously.

• Events T & F are not independent, as event F ie getting first white ball effects event T ie getting the white ball twice.
• Events F & S are not mutually exclusive, as event F ie getting first white ball & event S picking the second white ball - can happen together
• Events F & S are independent, as event F ie getting first white ball & event S picking the second white ball - do not effect each other.