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Difficulty: Very hardExperimental and Theoretical Probability

A biased four-sector spinner has sectors labeled 1, 2, 3, and 4. The theoretical probabilities of landing on sectors 1, 2, and 3 are in the ratio 1:2:21 : 2 : 2, respectively, while the theoretical probability of landing on sector 4 is 0.200.20. In an experiment consisting of 200200 spins, sector 2 was recorded 8080 times and sector 4 was recorded 6060 times. What is the positive difference between the expected number of even outcomes in 500500 future spins based on the experimental probability and that based on the theoretical probability?

  1. 9090Answer
  2. B
    5050
  3. C
    318318
  4. D
    6060

Answer

90
The theoretical probability of getting an even outcome (sector 2 or 4) is P(2)+P(4)=0.32+0.20=0.52P(2) + P(4) = 0.32 + 0.20 = 0.52, which predicts 500×0.52=260500 \times 0.52 = 260 even outcomes in 500500 spins. Empirically, even outcomes occurred 80+60=14080 + 60 = 140 times in 200200 spins, giving an experimental probability of 140/200=0.70140 / 200 = 0.70 and an estimated frequency of 500×0.70=350500 \times 0.70 = 350. The positive difference between these two expected values is 350−260=90350 - 260 = 90.

Step-by-Step Solution

1
Calculate the theoretical probabilities for each sector
P(1)=0.16P(1) = 0.16, P(2)=0.32P(2) = 0.32, P(3)=0.32P(3) = 0.32, P(4)=0.20P(4) = 0.20
Since P(4)=0.20P(4) = 0.20, the remaining probability 1−0.20=0.801 - 0.20 = 0.80 is shared among sectors 1, 2, and 3 in the ratio 1:2:21 : 2 : 2 (total 5 parts). Each part equals 0.80/5=0.160.80 / 5 = 0.16.
2
Determine the theoretical probability and expected frequency of an even outcome
Ptheo(Even)=0.52P_{\text{theo}}(\text{Even}) = 0.52; Expected frequency in 500 spins =260= 260
The even outcomes are sectors 2 and 4. Ptheo(Even)=P(2)+P(4)=0.32+0.20=0.52P_{\text{theo}}(\text{Even}) = P(2) + P(4) = 0.32 + 0.20 = 0.52. In 500500 spins, 500×0.52=260500 \times 0.52 = 260.
3
Calculate the experimental probability and expected frequency of an even outcome
Pexp(Even)=0.70P_{\text{exp}}(\text{Even}) = 0.70; Expected frequency in 500 spins =350= 350
Out of 200200 spins, even outcomes occurred 80+60=14080 + 60 = 140 times. Pexp(Even)=140/200=0.70P_{\text{exp}}(\text{Even}) = 140 / 200 = 0.70. In 500500 spins, 500×0.70=350500 \times 0.70 = 350.
4
Find the positive difference between the two expected values
350−260=90350 - 260 = 90
Subtracting the theoretical expected frequency from the experimental expected frequency gives ∣350−260∣=90|350 - 260| = 90.

Key Concept

Experimental vs. Theoretical Probability and Expected Frequency
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Topics

Experimental and Theoretical Probability

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