probability practice question 1

Probability Analysis of Student Performance Based on Gender

In a group of 600 students, there are 400 females and 200 males. Among these, 100 students have failed the exam, and 40 of them are male. Below is a tabular representation of the data:

Gender	Failed (x)	Not Failed (x’)	Total
Female (F)	40	360	400
Male (F’)	60	140	200
Total	100	500	600

Probability Calculations

Basic Probabilities

• P(F): 400/600 = 2/3
• P(F’): 200/600 = 1/3
• P(x): 100/600 = 1/6
• P(x’): 500/600 = 5/6

Intersections

• P(F ∩ x): 40/600 = 0.067
• P(F’ ∩ x): 60/600 = 0.1
• P(F ∩ x’): 360/600 = 0.6
• P(F’ ∩ x’): 140/600 = 0.233

Unions

• P(F ∪ x): P(F) + P(x) – P(F ∩ x)
  = 400/600 + 100/600 – 40/600 = 0.7666
• P(F’ ∪ x): P(F’) + P(x) – P(F’ ∩ x)
  = 200/600 + 100/600 – 60/600 = 0.400
• P(F ∪ x’): P(F) + P(x’) – P(F ∩ x’)
  = 400/600 + 500/600 – 360/600 = 0.900
• P(F’ ∪ x’): P(F’) + P(x’) – P(F’ ∩ x’)
  = 200/600 + 500/600 – 140/600 = 0.9333

Conditional Probabilities

• P(F | x): P(F ∩ x) / P(x)
  = 40/100 = 0.4000
• P(F’ | x): P(F’ ∩ x) / P(x)
  = 60/100 = 0.6000
• P(F | x’): P(F ∩ x’) / P(x’)
  = 360/500 = 0.7200
• P(F’ | x’): P(F’ ∩ x’) / P(x’)
  = 140/500 = 0.2800
• P(x | F): P(F ∩ x) / P(F)
  = 40/400 = 0.1000
• P(x’ | F): P(F ∩ x’) / P(F)
  = 360/400 = 0.9000
• P(x | F’): P(F’ ∩ x) / P(F’)
  = 60/200 = 0.3000
• P(x’ | F’): P(F’ ∩ x’) / P(F’)
  = 140/200 = 0.7000

Conclusion

From the calculations above, we can observe key trends in student performance based on gender. The probability of failing is higher for males (30%) than for females (10%). Additionally, females have a higher probability of passing (90%) compared to males (70%). This data can be useful for further analysis of educational trends and intervention strategies.