Exercise 7.2 Practice
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Overview
This page provides comprehensive Ch 7: The Mathematics of Maybe: Introduction to Probability - Exercise Set 7.2 Practice. Practice experimental probability sweeps, representative club survey proportions, coin flips, die trials, paper cup asymmetric landing, and Law of Large Numbers checks with step-by-step solutions.
Experimental Probability, Sweets Sampling, Surveys & Physical Probability Trials
Q1: Sweet Bag Sampling
A teacher mixes a large bag of sweets of different colours and randomly selects a sample of $30\text{ sweets}$. She counts:
• **10 red sweets** | **8 green sweets** | **7 yellow sweets** | **5 blue sweets**
(i) Calculate the probability that a randomly picked sweet from the sample is green.
(ii) If there are $600\text{ sweets}$ in total in the large bag, estimate how many are likely to be yellow, based on the sample results.
• **10 red sweets** | **8 green sweets** | **7 yellow sweets** | **5 blue sweets**
(i) Calculate the probability that a randomly picked sweet from the sample is green.
(ii) If there are $600\text{ sweets}$ in total in the large bag, estimate how many are likely to be yellow, based on the sample results.
(i) P(green):
Total sweets in sample $N = 30$. Green sweets count $= 8$.
$$P(\text{Green}) = \frac{8}{30} = \mathbf{\frac{4}{15} \approx 0.267}$$
Total sweets in sample $N = 30$. Green sweets count $= 8$.
$$P(\text{Green}) = \frac{8}{30} = \mathbf{\frac{4}{15} \approx 0.267}$$
(ii) Estimate Yellow sweets:
Sample probability for yellow sweet:
$$P(\text{Yellow}) = \frac{7}{30}$$
Project to the total bag size $M = 600$:
$$\text{Estimated Yellow} = P(\text{Yellow}) \times M = \frac{7}{30} \times 600 = 7 \times 20 = \mathbf{140\text{ sweets}}$$
Sample probability for yellow sweet:
$$P(\text{Yellow}) = \frac{7}{30}$$
Project to the total bag size $M = 600$:
$$\text{Estimated Yellow} = P(\text{Yellow}) \times M = \frac{7}{30} \times 600 = 7 \times 20 = \mathbf{140\text{ sweets}}$$
(i) 4/15 (0.267) (ii) 140 yellow sweets
Q2: School Club Survey Estimate
A survey is conducted at a school where a random sample of $40\text{ students}$ is asked about their favourite club:
• **14 students:** Science Club | **11 students:** Arts Club | **9 students:** Sports Club | **6 students:** Debate Club
Assume there are $800\text{ students}$ in the whole school.
(i) What is the probability that a randomly chosen student from the sample prefers the Arts Club?
(ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.
• **14 students:** Science Club | **11 students:** Arts Club | **9 students:** Sports Club | **6 students:** Debate Club
Assume there are $800\text{ students}$ in the whole school.
(i) What is the probability that a randomly chosen student from the sample prefers the Arts Club?
(ii) Using the sample results, estimate how many students in the whole school are likely to prefer the Sports Club.
(i) Arts Club Probability:
Sample size $N = 40$. Arts club count $= 11$.
$$P(\text{Arts}) = \frac{11}{40} = \mathbf{0.275 \text{ (or 27.5\%)}}$$
Sample size $N = 40$. Arts club count $= 11$.
$$P(\text{Arts}) = \frac{11}{40} = \mathbf{0.275 \text{ (or 27.5\%)}}$$
(ii) Estimate Sports Club preference:
Sample probability for Sports club:
$$P(\text{Sports}) = \frac{9}{40}$$
Project to school population $M = 800$:
$$\text{Estimated Students} = \frac{9}{40} \times 800 = 9 \times 20 = \mathbf{180\text{ students}}$$
Sample probability for Sports club:
$$P(\text{Sports}) = \frac{9}{40}$$
Project to school population $M = 800$:
$$\text{Estimated Students} = \frac{9}{40} \times 800 = 9 \times 20 = \mathbf{180\text{ students}}$$
(i) 0.275 (ii) 180 students
Q3: 20-Coin Toss Experiment
Toss a coin $20\text{ times}$ and record the result each time (heads or tails).
(i) How many times did you get heads?
(ii) How many times did you get tails?
(iii) Calculate the experimental probability of getting heads.
(iv) If you toss the coin once more, what is the probability of getting tails?
(i) How many times did you get heads?
(ii) How many times did you get tails?
(iii) Calculate the experimental probability of getting heads.
(iv) If you toss the coin once more, what is the probability of getting tails?
(i) & (ii) Sample Experimental Counts:
Suppose in your trial of 20 flips, you obtain:
• Heads count = **$11$**
• Tails count = **$9$**
Suppose in your trial of 20 flips, you obtain:
• Heads count = **$11$**
• Tails count = **$9$**
(iii) Experimental Probability:
$$P_{exp}(\text{Heads}) = \frac{\text{Heads count}}{20} = \frac{11}{20} = \mathbf{0.55}$$
$$P_{exp}(\text{Heads}) = \frac{\text{Heads count}}{20} = \frac{11}{20} = \mathbf{0.55}$$
(iv) Next Toss Probability:
Since each toss is an independent event, the coin has no memory of the previous 20 tosses. The theoretical probability of getting tails remains exactly **$1/2 = 50\% = 0.50$**.
Since each toss is an independent event, the coin has no memory of the previous 20 tosses. The theoretical probability of getting tails remains exactly **$1/2 = 50\% = 0.50$**.
(i) 11 (sample) (ii) 9 (sample) (iii) 0.55 (sample) (iv) 0.50
Q4: Paper Cup Toss Experiment
Toss a paper cup into the air $100\text{ times}$. After each toss record whether the cup lands on its bottom, upside down on its top, or on its side. Assign probabilities to the outcomes by using experimental probability.
Model Trials Data:
Since a paper cup is an asymmetric 3D shape, its landing options are not equally likely.
Suppose in 100 trials, the outcomes are:
• **Lands on Bottom:** 15 times
• **Lands on Top (Upside Down):** 25 times
• **Lands on Side:** 60 times
Since a paper cup is an asymmetric 3D shape, its landing options are not equally likely.
Suppose in 100 trials, the outcomes are:
• **Lands on Bottom:** 15 times
• **Lands on Top (Upside Down):** 25 times
• **Lands on Side:** 60 times
Experimental Probability Assignment:
$$P(\text{Bottom}) = \frac{15}{100} = \mathbf{0.15}$$
$$P(\text{Top}) = \frac{25}{100} = \mathbf{0.25}$$
$$P(\text{Side}) = \frac{60}{100} = \mathbf{0.60}$$
$$P(\text{Bottom}) = \frac{15}{100} = \mathbf{0.15}$$
$$P(\text{Top}) = \frac{25}{100} = \mathbf{0.25}$$
$$P(\text{Side}) = \frac{60}{100} = \mathbf{0.60}$$
Sample Probabilities: P(Bottom) = 0.15, P(Top) = 0.25, P(Side) = 0.60
Q5: Die Even Number
What is the probability of getting an even number when rolling a fair 6-sided die?
Sample Space $S = \{1, 2, 3, 4, 5, 6\} \implies n(S) = 6$.
Favourable even outcomes $E = \{2, 4, 6\} \implies n(E) = 3$.
$$P(\text{Even}) = \frac{n(E)}{n(S)} = \frac{3}{6} = \mathbf{0.50 \text{ (or 50\%)}}$$
Favourable even outcomes $E = \{2, 4, 6\} \implies n(E) = 3$.
$$P(\text{Even}) = \frac{n(E)}{n(S)} = \frac{3}{6} = \mathbf{0.50 \text{ (or 50\%)}}$$
P(Even) = 0.50
Q6: Die Trial Analysis
Suppose you roll a 6-sided die 12 times and get a ‘3’ three times.
(i) What is the experimental probability of rolling a ‘3’?
(ii) What is the theoretical probability of rolling a ‘3’?
(iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?
(i) What is the experimental probability of rolling a ‘3’?
(ii) What is the theoretical probability of rolling a ‘3’?
(iii) Why might these probabilities be different? What would you expect to happen if you roll the die 60, 600, or 6000 times?
(i) Experimental Probability:
$$P_{exp}(3) = \frac{3}{12} = \mathbf{0.25}$$
$$P_{exp}(3) = \frac{3}{12} = \mathbf{0.25}$$
(ii) Theoretical Probability:
$$P_{th}(3) = \mathbf{\frac{1}{6} \approx 0.167}$$
$$P_{th}(3) = \mathbf{\frac{1}{6} \approx 0.167}$$
(iii) Convergence explanation:
The difference is due to the small sample size ($N = 12$). In short runs, random variations can significantly affect relative frequencies.
As the number of rolls increases to 60, 600, or 6000, by the Law of Large Numbers, the experimental probability will stabilize and converge closer and closer to the theoretical value of $1/6 \approx 16.67\%$.
The difference is due to the small sample size ($N = 12$). In short runs, random variations can significantly affect relative frequencies.
As the number of rolls increases to 60, 600, or 6000, by the Law of Large Numbers, the experimental probability will stabilize and converge closer and closer to the theoretical value of $1/6 \approx 16.67\%$.
(i) 0.25 (ii) 0.167 (iii) Convergence to 1/6 as trials increase