A bin contains 4 apples and 3 oranges. 2 pieces of fruit are selected at random without replacement.

Question

a) Use a tree diagram to construct a probabilistic model for this experiment.

b) What is the probability that the sample contains at least one orange?

c) What is the probability that they are different types of fruit?

d) What is the probability that the second piece of fruit is an orange?

e) What is the probability that the second one is an orange given that the two pieces of fruit are different types?

f) Are the events in c) and d) independent?

g) What is the expected number of oranges?

Answer 1

On the 1st draw we have P(A) = 4/7 and P(O) = 3/7 

 

If the 1st draw is A then on the 2nd draw we have P(A) = 3/6 and P(O) = 3/6

If the 1st draw is O then on the 2nd draw we have P(A) = 4/6 and P(O) = 2/6  

 

P(AA) = (4/7)*(3/6) = 12/42 = 2/7

P(AO) = (4/7)*(3/6) = 12/42 = 2/7

P(OA) = (3/7)*(4/6) = 12/42 = 2/7

P(OO) = (3/7)*(2/6) =  6/42  = 1/7

 

P(O>=1) = 1 - P(AA) = 1 - 2/7 = 5/7 

 

P(different) = P(AO) + P(OA) = 2/7 + 2/7 = 4/7

 

P(draw 2 = O) = P(AO) + P(OO) = 2/7 + 1/7 = 3/7

 

P(draw 2=O | different) = (2/7) / (4/7) = 1/2 

 

c) and d) are not independent.  If they were then d) and e) would have the same probability 

 

E(#O) = (2/7)*0 + (2/7 + 2/7)*1 + (1/7)*2 = 5/7