Tree Diagrams in Probability
Senior 2 · Mathematics
Tree Diagrams in Probability
Lesson 4 of 4
22 min
"Add" or "multiply"? Whether you may simply add or multiply two probabilities depends on how the events are related. Two ideas settle it: whether the events can happen together (mutually exclusive), and whether one affects the other (independent). Getting them confused is one of the most common mistakes in probability examinations.
Events A and B are mutually exclusive if they cannot happen at the same time: they have no outcome in common, so P(A and B) = 0. For mutually exclusive events:
P(A or B) = P(A) + P(B)
Worked example 1
A die is thrown. A = {1, 2} and B = {5}. They cannot both happen, so P(A or B) = 26 + 16 = 36 = 12.
Worked example 2
(not mutually exclusive). A = "even" = {2, 4, 6} and B = "greater than 3" = {4, 5, 6}. Both contain 4 and 6. Adding gives 36 + 36 = 1, which is wrong because 4 and 6 are counted twice. Counting directly, A or B = {2, 4, 5, 6}, so P(A or B) = 46 = 23.
Events A and B are independent if the outcome of one does not change the probability of the other. For independent events:
P(A and B) = P(A) × P(B)
This is the "multiply along the branches" rule, in the case where the second-stage branches do not depend on the first stage.
Worked example 3
A penalty taker scores with probability 0.8 each time, independently. P(scores twice) = 0.8 × 0.8 = 0.64.
Worked example 4
(testing for independence). A die is thrown. A = "even" and B = "a multiple of 3". P(A) = 12, P(B) = 26 = 13, and A and B = {6}, so P(A and B) = 16. Since 12 × 13 = 16, A and B are independent.
Mutually exclusive events with non-zero probabilities are never independent: if A happens, B certainly cannot, so A changes the probability of B to 0.
Tip
OR with no overlap: add. AND with no influence: multiply. Before adding, ask "can both happen at once?". Before multiplying, ask "does the first change the second?".
Cards numbered 1 to 10: find P(a prime or the number 10).
Solution: primes {2, 3, 5, 7} and {10} do not overlap: 410 + 110 = 510 = 12.
P(A) = 0.3 and P(B) = 0.5, and A and B are mutually exclusive. Find P(A or B).
Solution: 0.3 + 0.5 = 0.8.
P(A) = 0.3 and P(B) = 0.5, and A and B are independent. Find P(A and B).
Solution: 0.3 × 0.5 = 0.15.
P(A) = 0.4, P(B) = 0.5 and P(A and B) = 0.2. Are A and B independent? Mutually exclusive?
Solution: 0.4 × 0.5 = 0.2, so they are independent. P(A and B) is not 0, so they are not mutually exclusive.
Two fair dice are thrown. Find P(double six).
Solution: the dice are independent: 16 × 16 = 136.
Remember
Sources and credits
The Rwandan examples, practice items and exercises are written by Ganzaa.
