Practice Maths

Sample Spaces — Solutions

  1. The Likelihood Scale

    1. Order: Impossible → Unlikely → Equally Likely → Likely → Highly Likely → Certain ▶ View Solution
    2. Rolling a 7 on a standard die: Impossible ▶ View Solution
    3. Flipping heads: Equally Likely ▶ View Solution
    4. Drawing a red card from a deck: Equally Likely ▶ View Solution
    5. Rolling less than 7: Certain ▶ View Solution

  2. Sample Spaces

    1. Coin: {Heads, Tails} ▶ View Solution
    2. Weekend day: {Saturday, Sunday} ▶ View Solution
    3. Die: {1, 2, 3, 4, 5, 6} ▶ View Solution
    4. Vowels in “MATHS”: {A} ▶ View Solution
    5. Coloured spinner: {Red, Blue, Green, Yellow} ▶ View Solution

  3. The Marble Bag

    1. Total marbles: 10 ▶ View Solution
    2. P(red): 25 ▶ View Solution
    3. P(green): 25 ▶ View Solution
    4. P(white): 15 ▶ View Solution
    5. P(blue): 0 ▶ View Solution

  4. Fractions, Decimals, Percentages

    1. P(red): 12,   0.5,   50% ▶ View Solution
    2. P(blue): 310,   0.3,   30% ▶ View Solution
    3. P(green): 15,   0.2,   20% ▶ View Solution
    4. Do all probabilities sum to 1? Yes — all probabilities in a sample space sum to 1 ▶ View Solution
    5. P(not blue): 0.7 ▶ View Solution

  5. Complementary Events

    1. P(winning): 310 ▶ View Solution
    2. P(not winning): 710 ▶ View Solution
    3. Do they sum to 1? Yes — complementary events always sum to 1 ▶ View Solution
    4. P(not winning) if P(winning) = 0.35: 0.65 ▶ View Solution

  6. Equally Likely Outcomes

    1. Fair die: Yes ▶ View Solution
    2. Bag with 3 red and 8 blue: No ▶ View Solution
    3. Unequal spinner sections: No ▶ View Solution
    4. Example of equal and not equal: Equal: rolling a fair die. Not equal: bag with 10 red and 1 blue ball. ▶ View Solution

  7. Spinner Logic

    1. P(purple): 14 ▶ View Solution
    2. As decimal and percentage: 0.25 and 25% ▶ View Solution
    3. P(not purple): 34 ▶ View Solution
    4. Expected purple in 80 spins: 20 times ▶ View Solution

  8. Real-World Probability

    1. P(rain) = 3 out of 4: 34,   75% ▶ View Solution
    2. P(bus on time) = 1 in 5: 15,   0.2,   20% ▶ View Solution
    3. Order least to most likely (13, 45%, 0.75): 13,   45%,   0.75 ▶ View Solution
    4. Is P = 1.3 valid? No — probability must be between 0 and 1 inclusive ▶ View Solution

  9. Trials, Outcomes and Events

    1. Sample space: {1, 2, 3, 4, 5, 6} ▶ View Solution
    2. P(rolling a 4): 16 ▶ View Solution
    3. Experimental P(4) after 30 rolls getting 4 eight times: 415 ▶ View Solution
    4. Is this evidence the die is biased? No — 30 trials is too few to judge; results naturally vary from theory by chance ▶ View Solution

  10. The Spinner Game

    1. Probabilities: P(red) = 25   P(blue) = 14   P(green) = 15   P(yellow) = 320 ▶ View Solution
    2. Experimental P(red) after 40 spins getting 14 reds: 720 (0.35) — below theoretical (0.40) ▶ View Solution
    3. Expected counts in 20 spins: Red: 8   Blue: 5   Green: 4   Yellow: 3 ▶ View Solution