Finding Probability Using Tree Diagrams or Tables

Can you list out all the combinations of food and drink? Ham macaroni + Soft drink, Ham macaroni + Tea, Ham macaroni + Coffee, ⋮ There are too many combinations. I am afraid I will miss some of them.

You can list them out effectively by drawing a tree diagram. For each kind of food, there are 4 choices for drink. Write down the choices in the 2nd column for each kind of food. There are 3 choices for food. Write down the choices in the 1st column. List out the combinations.

12 different combinations If we assume that the food and drink are randomly selected, each combination will occur with a probability of .

Let’s study a simple example. For two true-or-false questions, the answers for the first and second questions are ‘true’ and ‘false’ respectively. If Albert chooses his answers at random, find the probabilities that he gets (a) both answers correct, (b) only one answer correct. Let’s study a simple example. Let T stand for true and F stand for false. 1st question T F 2nd question F T Possible Outcomes T, T T, F F, T F, F 4 possible outcomes From the tree diagram, there are 4 equally likely outcomes. (a) There is only 1 favourable outcome, i.e. T, F. 1 ∴ P(both answers are correct) 4

For two true-or-false questions, the answers for the first and second questions are ‘true’ and ‘false’ respectively. If Albert chooses his answers at random, find the probabilities that he gets (a) both answers correct, (b) only one answer correct. Let T stand for true and F stand for false. 1st question T F 2nd question F T Possible Outcomes T, T T, F F, T F, F 4 possible outcomes From the tree diagram, there are 4 equally likely outcomes. (b) There are 2 favourable outcomes, i.e. T, T and F, F. 4 2 ∴ P(only one answer is correct)

Follow-up question Solution Mr Li has 3 children. Find the probability that only 2 of them are girls. Solution Let B stand for a boy and G stand for a girl. 1st child B G 2nd child G B G 3rd child B Possible Outcomes B, B, B B, B, G B, G, B B, G, G G, B, B G, B, G G, G, B G, G, G 8 possible outcomes in which 3 outcomes have two daughters. From the tree diagram, there are 8 equally likely outcomes 3 ∴ P(only 2 of the children are girls) 8

Let H stand for a head and T stand for a tail. Besides using tree diagrams, we may also list all the possible outcomes in tables. However, it is much simpler to use tree diagrams for 3 or more successive events. For example, the table below lists all the possible outcomes of tossing a coin twice. Let H stand for a head and T stand for a tail. possible outcomes of the first toss Second toss  H T  possible outcomes of the second toss First toss H HH HT 4 possible outcomes T TH TT

Follow-up question Tom chooses a letter at random from each of the two words ‘SKY’ and ‘SHY’. List all the possible outcomes in a table. Hence, find the probability that the two letters are the same. Solution 2nd word S H Y S SS SH SY 9 possible outcomes 1st word K KS KH KY Y YS YH YY From the table, there are 9 possible outcomes in which 2 outcomes have the same letter. 2 ∴ P(two letters are the same) 9