For multiple trials, such as tossing a coin or rolling a die, write a tree diagram to count all possible outcomes without omissions.
Example ①
When a coin is tossed twice, find the probability that heads appears both times.
Solution: Tree diagram: (heads,heads)(heads,tails)(tails,heads)(tails,tails) → 4 outcomes in all. Heads appears both times in 1 outcome. P = 1/4
When a coin is tossed twice, find the probability that heads appears both times.
Solution: Tree diagram: (heads,heads)(heads,tails)(tails,heads)(tails,tails) → 4 outcomes in all. Heads appears both times in 1 outcome. P = 1/4
Example ②
When a coin is tossed three times, find the probability that heads appears at least once.
Solution: In all, 2³=8 outcomes. There is 1 outcome with tails every time. Heads appears at least once in 8−1=7 outcomes. P = 7/8
When a coin is tossed three times, find the probability that heads appears at least once.
Solution: In all, 2³=8 outcomes. There is 1 outcome with tails every time. Heads appears at least once in 8−1=7 outcomes. P = 7/8
Key Points
- Count all outcomes without omissions using a tree diagram
- For “at least once,” using the complementary event (“does not occur even once”) is faster
Practice Problems
- When a coin is tossed twice, find the probability that heads appears at least once.
- When two dice are rolled at the same time, find the probability that the sum of the numbers is 3.
Answers and Explanations
- Answer: 3/4. There are 4 outcomes in all, with 1 outcome consisting entirely of tails. 4−1=3 outcomes. 3/4.
- Answer: 1/18. There are 36 outcomes in all. A sum of 3 occurs in (1,2)(2,1), or 2 outcomes. 2/36=1/18.