Let us say that you have a box containing 3 red pens, 4 black pens and 2 green pens. You pick a pen (without looking) from the box and put it back. Then your friend does the same.
What are the possible outcomes of the pen colours? Can you draw a tree diagram representing the possible outcomes?
The possible outcomes of the pen colours are the 9 pairs: (Red, Red), (Red, Black), (Red, Green), (Black, Red), (Black, Black), (Black, Green), (Green, Red), (Green, Black), (Green, Green). Yes, a tree diagram can be drawn where the first set of branches shows my pick (Red, Black, Green) and the second set shows my friend's pick.
Can you use the tree diagram to guess the probability that both you and your friend pick pens of the same colour?
Yes, the tree diagram can be used to guess the probability. The favourable outcomes for picking the same colour are (Red, Red), (Black, Black), and (Green, Green). Using the branches, P(Red, Red) = 1/9, P(Black, Black) = 16/81, and P(Green, Green) = 4/81. Adding these gives a total probability of 29/81.
Explanation
The student identifies the sample space for two independent draws with replacement and visualizes it using a tree diagram. Then, they use the tree diagram's branches to calculate the probability of the compound event (both picking the same colour) by multiplying probabilities along the specific branches and summing them up.