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Hard probability, permutations, combinations and statistical analysis!
1. What is the probability of getting at least one head when flipping 3 coins?
💡 P(at least one head) = 1 - P(no heads) = 1 - (1/2)³ = 1 - 1/8 = 7/8.
2. What is the harmonic mean of 2 and 8?
💡 Harmonic mean = 2/(1/2 + 1/8) = 2/(5/8) = 16/5 = 3.2.
3. What is 8P2 (permutation)?
💡 P(8,2) = 8!/(8-2)! = 8×7 = 56.
4. A card is drawn from a standard deck. What is P(King or Heart)?
💡 P(King) = 4/52, P(Heart) = 13/52, P(King of Hearts) = 1/52. P = 4+13-1/52 = 16/52. Actually 16/52 simplifies, so answer is C: 16/52.
5. How many diagonals does a hexagon have?
💡 Diagonals = n(n-3)/2 = 6(3)/2 = 9.
6. Two dice are rolled. What is the probability of getting a sum of 7?
💡 Combinations summing to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6. P = 6/36 = 1/6.
7. What does a p-value less than 0.05 indicate in hypothesis testing?
💡 A p-value < 0.05 means there is statistically significant evidence to reject the null hypothesis.
8. What is the geometric mean of 4 and 16?
💡 Geometric mean = √(4 × 16) = √64 = 8.
9. In a normal distribution, what percentage of data falls within 2 standard deviations of the mean?
💡 About 95% of data falls within 2 standard deviations of the mean in a normal distribution.
10. How many ways can 5 people sit in a row?
💡 5! = 5 × 4 × 3 × 2 × 1 = 120 ways.
11. What is 6C2 (combination)?
💡 C(6,2) = 6!/(2!×4!) = (6×5)/(2×1) = 15.
12. What is Bayes' theorem used for?
💡 Bayes' theorem calculates conditional probability, updating the probability of a hypothesis given new evidence.
13. How many 3-digit numbers can be formed using digits 1-9 without repetition?
💡 P(9,3) = 9 × 8 × 7 = 504.
14. A box has 4 red and 6 blue balls. Two are drawn without replacement. What is P(both red)?
💡 P = (4/10) × (3/9) = 12/90 = 2/15.
15. What does a correlation coefficient of -1 indicate?
💡 A correlation of -1 means a perfect negative linear relationship — as one variable increases, the other decreases perfectly.
16. What is the median of a normal distribution relative to its mean?
💡 In a perfect normal distribution, the mean, median, and mode are all equal.
17. What is the expected value of rolling a fair six-sided die?
💡 E = (1+2+3+4+5+6)/6 = 21/6 = 3.5.
18. What is the variance if the standard deviation is 6?
💡 Variance = SD² = 6² = 36.
19. In how many ways can the letters of MATHS be arranged?
💡 MATHS has 5 distinct letters. 5! = 120 arrangements.
20. What is the standard deviation of: 2, 4, 4, 4, 5, 5, 7, 9?
💡 Mean=5. Variance = average of squared deviations = 4. SD = √4 = 2.