Practice Maths

Investigating Probabilities — Solutions

  1. Definitions Check

    1. Calculated from outcomes without experimenting: Theoretical probability ▶ View Solution
    2. Based on actual results of trials: Experimental probability ▶ View Solution
    3. Frequency of an outcome divided by total trials: Relative frequency ▶ View Solution
    4. More trials bring experimental closer to theoretical: Law of Large Numbers ▶ View Solution
    5. One experiment or observation: A trial ▶ View Solution

  2. The 5-Section Spinner

    1. P(each colour): 15,   0.2,   20% ▶ View Solution
    2. Expected count per colour in 100 spins: 20 times each ▶ View Solution
    3. Experimental P(red) from 12 reds in 50 spins: 0.24 ▶ View Solution
    4. Higher or lower than theoretical? Higher ▶ View Solution
    5. With many more trials, what happens? Experimental probability gets closer to theoretical (Law of Large Numbers) ▶ View Solution

  3. Analysing Trial Size

    1. Theoretical P(heads): 50% ▶ View Solution
    2. Which trial size gave closest result (10 = 60%, 100 = 47%, 1000 = 50.4%)? 1 000 trials ▶ View Solution
    3. Why do more trials give more reliable results? More trials reduce the effect of random variation ▶ View Solution
    4. What does the Law of Large Numbers predict? As trials increase, experimental probability approaches theoretical probability ▶ View Solution
    5. Can exactly 50% ever occur in practice? Yes, possible but not guaranteed — it happens by chance ▶ View Solution

  4. Predicting Future Events

    1. Experimental P(goal) from 12 goals in 30 shots: 25 ▶ View Solution
    2. As decimal and percentage: 0.4,   40% ▶ View Solution
    3. Predicted goals in next 50 shots: 20 goals ▶ View Solution
    4. Updated probability after 18 goals in next 50: 925 ▶ View Solution
    5. Has the estimated probability gone up or down? Down — from 0.40 to 0.36 ▶ View Solution

  5. The Points Race

    1. P(odd on a die): 12 ▶ View Solution
    2. P(rolling a 6): 16 ▶ View Solution
    3. Who scores more often? Player A — P(A scores) = 0.5 vs P(B scores) ≈ 0.167 ▶ View Solution
    4. Is the game fair? No — Player A averages 2.5 pts per roll vs Player B ≈ 1.67 pts per roll ▶ View Solution

  6. Investigating the Marble Bag

    1. P(red) from bag with 3 red and 7 other: 0.3 ▶ View Solution
    2. Experimental P(red) from 4 reds in 20 draws: 0.2 ▶ View Solution
    3. Above or below theoretical? Below ▶ View Solution
    4. With 200 draws, experimental probability would be: Closer to 0.3 ▶ View Solution

  7. Equally Likely or Not?

    1. Rolling a fair die: Yes — each face has P = 16 ▶ View Solution
    2. Next car colour is red or not red: No — most cars are not red ▶ View Solution
    3. Give your own example of equally likely outcomes: e.g. flipping a fair coin: P(heads) = P(tails) = 12 ▶ View Solution
    4. Unequal sections spinner — which colour is most likely? The colour with the largest section ▶ View Solution

  8. Relative Frequency in Context

    1. Relative frequency from 40 late buses in 200 journeys: 15 ▶ View Solution
    2. As a percentage: 20% ▶ View Solution
    3. Predicted late buses in next 150 journeys: 30 buses ▶ View Solution
    4. Why is a larger sample more reliable? Reduces the effect of random variation, giving a better estimate of the true probability ▶ View Solution

  9. The Shape Bag Investigation

    1. P(square or circle) from bag of 20: 920 = 0.45 ▶ View Solution
    2. Experimental P from 26 in 50 draws: 0.52 ▶ View Solution
    3. Higher than theoretical? Does this suggest unfairness? Higher, but not strong evidence of unfairness — 50 trials naturally vary ▶ View Solution

  10. Investigating a Biased Die

    1. Expected sixes in 300 rolls: 50 sixes ▶ View Solution
    2. Experimental P from 85 sixes in 300: 28.3% ▶ View Solution
    3. Is the die likely biased? Yes — 28.3% is nearly double the expected 16.7%, and 300 trials is enough to suggest bias ▶ View Solution