Relative Frequency and Experimental Probability — Solutions

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1. Relative frequency from a table (die rolled 60 times)

   1. Relative frequencies: 1: 8/60 = 2/15; 2: 12/60 = 1/5; 3: 9/60 = 3/20; 4: 11/60; 5: 10/60 = 1/6; 6: 10/60 = 1/6
   2. Theoretical P(each): 1/6 ≈ 0.1667 for each outcome
   3. Highest relative frequency: Outcome 2 (12/60 = 0.2). Not necessarily surprising with only 60 trials — random variation is expected.
   4. Sum check: 8+12+9+11+10+10 = 60; 60/60 = 1 ✓

2. Experimental probability

   1. Coin: 28 heads in 50: P(heads) = 28/50 = 14/25 = 0.56 = 56%
   2. Drawing pin: 63 point up in 100: P(point up) = 63/100 = 0.63 = 63%
   3. Tennis: 52 successful in 80: P(success) = 52/80 = 13/20 = 0.65 = 65%
   4. Red lolly: 14 in 40 draws: P(red) = 14/40 = 7/20 = 0.35 = 35%
   5. Basketball: 18 in 25: P(score) = 18/25 = 0.72 = 72%

3. Expected frequency

   1. Rolling a 6 in 120 rolls: E = 1/6 × 120 = 20 times
   2. Tails in 300 flips: E = 1/2 × 300 = 150 tails
   3. Rainy days in June: E = 0.3 × 30 = 9 rainy days
   4. Spinner (2R, 2B, 1G) in 200 spins: P(red) = 2/5 → 80; P(blue) = 2/5 → 80; P(green) = 1/5 → 40
   5. 3R 7B bag, 50 draws: P(red) = 3/10; E = 3/10 × 50 = 15 reds expected

4. Compare experimental and theoretical

   1. Coin flipped 10 times (H,H,T,H,H,T,T,H,H,T):
      1. Heads = 6; P(heads) = 6/10 = 0.6
      2. Theoretical = 0.5; experimental = 0.6; difference of 0.1
      3. Closer. The law of large numbers states that experimental probability approaches theoretical as trials increase.
   2. Die rolled 30 times; 18 even:
      1. P(even) = 18/30 = 3/5 = 0.6
      2. Theoretical P(even) = 3/6 = 0.5
      3. Not necessarily biased. With only 30 trials, a difference of 0.1 can occur by chance. Many more trials (hundreds) would be needed to draw a reliable conclusion.

5. Law of large numbers and data

   1. Spinner results approaching 0.25: The pattern (0.4, 0.32, 0.253) suggests the theoretical P(red) ≈ 0.25. By the law of large numbers, as trials increase, experimental probability converges to theoretical probability.
   2. Factory defects: 8 in 200: P(defective) = 8/200 = 0.04 = 4%; In 5000: 0.04 × 5000 = 200 defective items expected
   3. Science preference: 30/50: P(science) = 30/50 = 3/5 = 0.6; With 500 students, the estimate would be more reliable — larger samples reduce the effect of random variation.
   4. 32 red in 80 draws: P(red) = 32/80 = 2/5; P(blue) = 3/5; Out of 10 marbles: 4 red and 6 blue

6. Problem solving

   1. Student late 6 in 40 days:
      1. P(late) = 6/40 = 3/20 = 0.15 = 15%
      2. Expected late days = 0.15 × 200 = 30 days
   2. Combined die results (A: 7 sixes in 30, B: 5 sixes in 30): Combined: 12 sixes in 60 rolls; P(6) = 12/60 = 1/5 = 0.2; Theoretical = 1/6 ≈ 0.167; Experimental is slightly higher but consistent with normal variation.
   3. Biased coin P(H) = 0.6, 500 flips: Expected heads = 0.6 × 500 = 300; Expected tails = 0.4 × 500 = 200; Difference = 100 more heads than tails.
   4. Conclusion about bias: With only 10 flips, getting 7 heads can easily happen by chance even with a fair coin (P ≈ 0.117). This is not strong evidence of bias. Better evidence would come from a much larger number of trials (e.g. 1000+) where the experimental probability consistently differs significantly from 0.5.

7. Design and interpret a spinner experiment

   1. 20-spin results: P(red)=4/20=0.2; P(blue)=9/20=0.45; P(green)=4/20=0.2; P(yellow)=3/20=0.15
   2. 200-spin results: P(red)=38/200=0.19; P(blue)=92/200=0.46; P(green)=41/200=0.205; P(yellow)=29/200=0.145
   3. 200-spin results are more reliable: The law of large numbers states that as the number of trials increases, experimental probability gets closer to the true theoretical probability. 200 spins provides a much more reliable estimate than 20 spins.
   4. Expected blue in 500 spins: P(blue)=92/200=0.46; Expected = 0.46 × 500 = 230 times.

8. Weather data — experimental probability

   1. Monthly experimental P(rain): January: 12/31 ≈ 0.387; February: 18/28 ≈ 0.643; March: 8/31 ≈ 0.258
   2. Combined P(rain): 38/90 ≈ 0.422 (42.2%)
   3. Predicted rainy days in a year: 0.422 × 365 ≈ 154 rainy days
   4. Limitation: Seasonal variation means rain probability differs by month (dry vs wet season). Using a three-month sample may not represent the whole year. Experimental probability is still a useful estimate but should be based on a full year of data for greater accuracy.

9. Quality control — light bulbs

   1. Fault rates: Batch A: P(faulty) = 15/500 = 0.03 = 3%; Batch B: P(faulty) = 32/800 = 0.04 = 4%. Batch B has the higher fault rate.
   2. Target check: Neither batch meets the 2% target — Batch A is 3% and Batch B is 4%, both exceeding 2%.
   3. Expected faulty in 10 000: Using Batch A rate: 0.03 × 10 000 = 300 faulty bulbs expected.
   4. New process: New fault rate = 8/1000 = 0.008 = 0.8%, which is well below the 2% target. Yes, the improvement has been effective.

10. Critical thinking — evaluating experimental probability

    1. Survey of 5 people: Problem 1: The sample size is far too small (only 5 people) to make a reliable generalisation about all consumers. Problem 2: The sample may not be representative — the 5 people may share similar preferences due to age, location, or other factors.
    2. Die rolled 600 times: Relative frequencies: 1→95/600≈0.158; 2→0.170; 3→0.165; 4→0.168; 5→0.163; 6→0.175. All are close to 1/6 ≈ 0.167. The die is likely fair — the variation is within normal range for 600 trials.
    3. Basketball coach’s claim: The claim is not reliable. 10 free throws is a very small sample; chance variation has a large effect. A better estimate would come from 100+ free throws. The coach should wait for more data before drawing conclusions about the player’s true ability.
    4. Pet ownership survey: Student B’s estimate should be trusted more. With 80 students, the sample is 10 times larger, reducing the effect of random variation. Student A’s estimate of 0.625 from only 8 people is far less reliable.