Trees/Combined Events

Use tree diagrams and combined-event reasoning to list outcomes and find probabilities.

All objectives

Foundation

16 web questions

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  1. Use this tree to list the possible outcomes for two fair coin tosses.
    Tree diagram for this question.
  2. Use the tree to say how many outcomes are possible when two fair coins are tossed.
  3. Use the tree to write the probability of getting HH.
  4. Use the tree to write the probability of getting TT.
  5. Use the tree to write the probability of getting exactly one head.
  6. Use this tree for a coin toss and a fair die roll. How many outcomes are possible altogether?
    Tree diagram for this question.
  7. Use the tree to find the probability of getting H and 3.
  8. Use the tree to find the probability of getting T and an even number.
  9. A spinner has equal sectors labelled red and blue. It is spun twice. List the possible colour outcomes.
  10. A spinner has equal sectors labelled red and blue. It is spun twice. What is the probability of red then blue?
  11. A bag has one red and one blue counter. A counter is picked, replaced, then picked again. List the possible colour outcomes.
  12. A bag has one red and one blue counter. A counter is picked, replaced, then picked again. What is the probability of getting two reds?
  13. Fill in the blank: when two fair coins are tossed, the probability of at least one tail is $\frac{\square}{4}$.
  14. Fill in the blank: when a coin is tossed and a die is rolled, the probability of heads is $\frac{\square}{12}$.
  15. Which is more likely when two fair coins are tossed: two heads or exactly one head?
  16. A student says there are only three outcomes when two coins are tossed: heads, tails, and one of each. Are they correct?

Proficient

18 web questions

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  1. Use this tree to list the full sample space for two fair coin tosses.
    Tree diagram for this question.
  2. Use the tree to find the probability of getting exactly two heads.
  3. Use the tree to find the probability of getting at least one head.
    Tree diagram for this question.
  4. Use this tree for a spinner labelled $1,2,3$ spun twice. List the outcomes where the total is $4$.
  5. Use the tree to find the probability that the total is $4$.
    Tree diagram for this question.
  6. A fair die is rolled twice. What is the probability of getting a 6 then an even number?
  7. A fair die is rolled twice. What is the probability of getting a 6 then an even number?
  8. A fair die is rolled twice. What is the probability that both numbers are odd?
  9. A bag has 2 red and 1 blue counter. A counter is picked, replaced, then picked again. What is the probability of red then blue?
  10. A bag has 2 red and 1 blue counter. A counter is picked, replaced, then picked again. What is the probability of two blues?
  11. A fair coin is tossed twice. What is the probability of getting the same result both times?
  12. A fair coin is tossed twice. What is the probability of getting different results?
  13. Fill in the blank: for two fair coin tosses, the probability of exactly one head is $\frac{\square}{4}$.
  14. Fill in the blank: for two fair die rolls, the probability of getting two odd numbers is $\frac{\square}{36}$.
  15. Which is greater: the probability of two heads in two coin tosses or the probability of a 6 then a 6 in two die rolls?
  16. Which is smaller: the probability of red then blue from a red-blue spinner spun twice or the probability of two reds?
  17. A student says the probability of H then 5 when a coin is tossed and a die is rolled is $\frac{1}{6}$. Are they correct?
  18. Explain in one short sentence what a branch on a tree diagram shows.

Excellence

19 web questions

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  1. Use this tree for Questions 1--4.
    Tree diagram for this question.
  2. Is the probability of at least one head equal to $\frac{1}{2}$? Explain.
  3. Use the tree to find the probability of getting exactly two heads.
  4. Use the tree to find the probability of getting at least one tail.
  5. A student says HTH and THH are the same because both have two heads and one tail. Are they correct? Explain using tree paths.
    Tree diagram for this question.
  6. Use this partial two-dice tree to help find the probability that the total is $7$.
  7. Use the tree to find the probability that both numbers are greater than 4.
  8. A spinner has equal sectors labelled A, B, and C. It is spun twice. What is the probability of getting exactly one A?
  9. A spinner has equal sectors labelled 1, 2, 3, and 4. It is spun twice. What is the probability that both spins are even?
  10. A bag has 3 red and 2 blue counters. A counter is picked, replaced, then picked again. What is the probability of getting one red and one blue in any order?
  11. A bag has 3 red and 2 blue counters. A counter is picked, replaced, then picked again. What is the probability of getting two counters of the same colour?
  12. Which is greater: the probability of exactly one head in three coin tosses or the probability of a total of 7 in two die rolls? Show enough working to justify.
  13. Which is smaller: the probability of two blues from a bag with 3 red and 2 blue counters, with replacement, or the probability of two heads in two coin tosses? Explain.
  14. Fill in the blank: when a fair coin is tossed three times, the probability of HHH is $\frac{\square}{8}$.
  15. Fill in the blank: when a fair die is rolled twice, the probability of getting doubles is $\frac{\square}{36}$.
  16. Which does not belong: outcome, branch, sample space, perimeter? Explain.
  17. A game pays out on exactly two heads from three fair coin tosses. Is this more or less likely than getting exactly one head? Explain.
  18. A coin is tossed and a die is rolled. Explain why the probability of H and an odd number is found by counting 3 favourable outcomes out of 12.
  19. Explain why a full tree diagram helps avoid missing combined outcomes.