The Case of the Missing Probability: Conditional Probability and Independence
In a certain school, 60% of students play sports, 40% play a musical instrument, and 30% do both. A student is selected at random. The school newspaper claims that playing sports and playing an instrument are independent events because the probability of playing an instrument given that the student plays sports is 0.5, which is close to the overall probability of playing an instrument (0.4). A statistics student argues that they are not independent because 0.5 ≠ 0.4. The newspaper editor counters that the difference is small and could be due to chance. The student decides to test independence using a formal definition and also considers a new scenario: if the school had 1000 students, how many would be expected to do both if the events were independent?
1. Using the given probabilities, determine whether playing sports and playing an instrument are independent. Show your reasoning using the formal definition of independence.
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Let S = plays sports, M = plays instrument. P(S)=0.6, P(M)=0.4, P(S∩M)=0.3. Events are independent if P(S∩M) = P(S)*P(M). Here P(S)P(M)=0.60.4=0.24. Since 0.3 ≠ 0.24, the events are not independent. Alternatively, P(M|S)=P(S∩M)/P(S)=0.3/0.6=0.5, which is not equal to P(M)=0.4. So dependence is confirmed.
2. If the school had 1000 students and the events were independent (with the same marginal probabilities), how many students would be expected to do both? How does this compare to the actual number based on the given 30%?
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If independent, P(S∩M)=0.60.4=0.24, so expected number = 0.241000=240. The actual number based on 30% is 0.3*1000=300. So there are 60 more students doing both than expected under independence. This indicates a positive association between the two activities.
3. The newspaper editor argues that the difference between 0.5 and 0.4 is small and might be due to sampling variability. If the sample size were 100 students instead of 1000, and the observed proportion doing both was 0.3, would the conclusion change? Discuss the role of sample size in testing independence.
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The conclusion about independence is a population parameter; sample size affects the precision of estimates. With n=100, the observed counts would be: sports=60, instrument=40, both=30. Expected both under independence = 24. The difference is 6. A chi-square test could be used, but the question asks about the conclusion. The sample size does not change the fact that in the population (as given by the probabilities) the events are not independent. However, with a smaller sample, the observed difference might not be statistically significant. But here the probabilities are given as population values, so independence is false regardless of sample size. The editor's argument confuses sample variability with population parameters. In hypothesis testing, with n=100, we would test H0: independence vs H1: dependence. The expected counts under H0: E11=24, E12=36, E21=16, E22=24. Chi-square = (30-24)^2/24 + (30-36)^2/36 + (10-16)^2/16 + (30-24)^2/24 = 36/24 + 36/36 + 36/16 + 36/24 = 1.5+1+2.25+1.5=6.25. df=1, p-value ≈ 0.012, so reject independence at 5% level. So even with n=100, we would likely detect dependence. But the key is that the conclusion about independence is about the population, not the sample.