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In an examination, 70% of the students who pass actually know the material, while the remaining 30% pass purely by guessing. A student who knows the material passes with probability 1, while a student who guesses passes with probability 0.25 (out of 4 options). It is known that 60% of all students who appear know the material, and the rest attempt by guessing. If a randomly selected student passes, what is the probability that the student actually knew the material?

In an examination, 70% of the students who pass actually know the material, while the remaining 30% pass purely by guessing. A student who knows the material passes with probability 1, while a student who guesses passes with probability 0.25 (out of 4 options). It is known that 60% of all students who appear know the material, and the rest attempt by guessing. If a randomly selected student passes, what is the probability that the student actually knew the material?

  1. 67\frac{6}{7}

  2. 37\frac{3}{7}

  3. 57\frac{5}{7}

  4. 47\frac{4}{7}

  5. 27\frac{2}{7}

Show answer & explanation

Correct answer

67\frac{6}{7}

Explanation

Let K = knows material, G = guesses. P(K)=0.6P(K)=0.6, P(G)=0.4P(G)=0.4. P(Pass∣K)=1P(\text{Pass}|K)=1, P(Pass∣G)=0.25P(\text{Pass}|G)=0.25. P(Pass)=0.6×1+0.4×0.25=0.6+0.1=0.7P(\text{Pass}) = 0.6\times1 + 0.4\times0.25 = 0.6+0.1=0.7. By Bayes' theorem, P(K∣Pass)=0.6×10.7=67P(K|\text{Pass}) = \frac{0.6\times1}{0.7} = \frac{6}{7}.

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All 30 Probability questions

  1. 1.A bag contains 4 red balls and 6 blue balls.
  2. 2.A die is rolled once.
  3. 3.A coin is tossed twice.
  4. 4.A card is drawn from a well-shuffled deck of 52 cards.
  5. 5.A box contains 3 red, 5 green, and 2 yellow balls.
  6. 6.What is the probability of getting a sum of 7 when two dice are rolled together?
  7. 7.In a class, there are 15 boys and 10 girls.
  8. 8.A spinner has 8 equal sections numbered 1 to 8.
  9. 9.A bag contains 5 red and 7 black balls.
  10. 10.If P(A) = 0.4 and P(B) = 0.5 for independent events A and B, what is P(A B) ?
  11. 11.A bag contains 4 red, 5 blue, and 6 green balls.
  12. 12.In a box, there are 3 defective and 7 non-defective items.
  13. 13.A card is drawn from a well-shuffled deck of 52 cards.
  14. 14.Two dice are rolled simultaneously.
  15. 15.A speaks truth in 75% cases and B speaks truth in 80% cases.
  16. 16.A bag contains 5 white and 3 black balls.
  17. 17.From a group of 6 men and 4 women, a committee of 3 persons is to be formed.
  18. 18.The probability that a student will pass in Mathematics is 0.6 and that in English is 0.5.
  19. 19.A die is thrown twice.
  20. 20.In a lottery, there are 5 prizes and 20 blanks.
  21. 21.A fair coin is tossed 5 times.
  22. 22.Two fair dice are rolled together.
  23. 23.Three fair dice are rolled simultaneously.
  24. 24.Two cards are drawn successively without replacement from a well-shuffled standard deck of 52 cards.
  25. 25.From a well-shuffled standard deck of 52 cards, 3 cards are drawn at random simultaneously.
  26. 26.An urn contains 6 white, 5 black and 4 red balls.
  27. 27.An urn contains 7 red and 5 blue balls.
  28. 28.For two events A and B, P(A) = 0.5 , P(B) = 0.4 and P(A B) = 0.2 .
  29. 29.For two events A and B, P(A) = 0.45 , P(B) = 0.35 and P(A B) = 0.15 .
  30. 30.In an examination, 70% of the students who pass actually know the material, while the remaining 30% pass purely by guessing.