5-INDEPENDENT EVENTS

The conditional probability of E given F, 

is not generally equal to P(E). In  other  words,  knowing  that  F  has occurred 

generally  changes  the chances of E’s occurrence. In the special cases where P(E|F)

does in fact  equal  P(E),  we  say  that  E  is  independent  of  F.

That  is,  E  is independent of F if knowledge that F has occurred does not change the probability

that E occurs. Since P(E|F) = P(E ∩ F)/P(F), it follows that

E is independent of F if P(E ∩ F)=P(E) P(F) By symmetric, whenever

E is independent of F, F is also independent of E. We thus have the following definition. 

Two events E and F are said to be independent 

if  P(E∩F) = P(E) P(F) So Two events E and F are said to be independent

if  P(E|F) = P(E) and P(F|E) = P(F) Two events E and F

that are not independent are said to be dependent.

A card is selected at random from an ordinary deck of 52 playing cards.

If E is the event that the selected card is an ace and F is the event

that it is a spade, then E and F are independent.

This follows because P(E ∩ F) =, whereas P(E) =  and P(F) = 

Two coins are flipped, and all 4 outcomes are assumed to be equally likely.

If E is the event that the first coin lands on heads and F the event that the second 

lands on  tails, then E and F are independent

, since P(E ∩ F) = P({(H, T)}) = whereas

P(E) = P({(H,H), (H, T)}) =  and P(F) = P({(H, T), (T, T)}) =  

Call a household prosperous if its income exceeds $100; 000. 

Call the household  educated  if  the  household  completed  college.

Select  an American household at random, and let A be the event that the

selected household  is  prosperous  and  let  B  be  the  event 

that  it  is  educated. According  to  the  Current  Population  Survey,

P(A) =  0.134,  P(B)  = 0.254, P(A|B) = 0.080. Are events A and B independent?