Topic Review — Probability

Review Questions

1. Tree diagrams and multi-step probability. Fluency

   1. A bag has 3 red and 2 blue marbles. One marble is drawn and replaced, then a second is drawn. Find P(both red).
   2. A fair coin is flipped twice. Find P(at least one Head).
   3. A box contains 4 green and 1 white ball. Two are drawn without replacement. Find P(both green).
   4. A spinner with equal sectors {A, B, C} is spun and a four-sided die is rolled. How many outcomes are in the sample space?

2. Venn diagrams — setting up and reading. Fluency

   1. In a class of 30: 18 have a mobile phone (M), 12 have a tablet (T), and 6 have both. Draw a Venn diagram and find P(has neither).
   2. Using the same data, find P(M ∪ T).
   3. P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2. Find P(A ∪ B).
   4. Are events A and B in (c) mutually exclusive? Explain.

3. Two-way tables. Understanding

   A survey of 80 teenagers asked about music preference and whether they play an instrument.

   	Pop	Rock	Other	Total
   Plays Instrument	14	18	8	40
   Does Not Play	22	10	8	40
   Total	36	28	16	80

   1. Find P(prefers Rock AND plays an instrument).
   2. Find P(plays an instrument | prefers Rock).
   3. Find P(prefers Pop OR plays an instrument).
   4. Are “plays instrument” and “prefers Rock” independent? Check using P(instrument ∩ Rock) = P(instrument) × P(Rock).

4. Experimental probability and fairness. Understanding

   1. A die is rolled 180 times. What is the expected frequency of rolling a 3?
   2. The number 3 was actually rolled 42 times. Find the experimental probability of rolling a 3.
   3. Compare the experimental and theoretical probabilities. Does the die appear biased?
   4. Explain why it would be more convincing to roll the die 1800 times rather than 180 times to test for bias.

5. Mixed problem solving. Problem Solving

   1. A class of 25 students: 10 study French (F), 8 study Japanese (J), 3 study both. A student is chosen at random.
      1. Draw a Venn diagram showing the numbers in each region.
      2. Find P(studies French or Japanese).
      3. Find P(studies Japanese given they study French).
   2. A weather app says P(rain tomorrow) = 0.6. James walks to school if it doesn’t rain (probability 0.3 of being late when walking) and gets a lift if it rains (probability 0.05 of being late with a lift).
      1. Draw a tree diagram showing all outcomes.
      2. Find P(James is late tomorrow).
      3. Given that James is late, find P(it was raining).
   3. A spinner is spun 500 times. The results are: Red 180, Blue 165, Green 155. A student says the spinner is fair because “the results are close enough to 1/3 each.” Do you agree? Justify your answer using both calculations and reasoning.