Statistics & Probability - Flashcards
Master Statistics & Probability concepts through practice problems and step-by-step solutions. These flashcards focus on problem-solving techniques essential for UPCAT and other college entrance examinations. Each card includes worked examples with detailed explanations to build your mathematical reasoning skills.
Flashcards
Find the mean of the data set: 85, 92, 78, 96, 88, 91, 84
Answer
Step 1: Add all values: 85 + 92 + 78 + 96 + 88 + 91 + 84 = 614
Step 2: Count the number of values: 7
Step 3: Divide sum by count: Mean = 614 ÷ 7 = 87.71
Answer: The mean is approximately 87.71
Find the median of: 15, 23, 12, 19, 27, 21, 18, 25, 16
Answer
Step 1: Arrange in ascending order: 12, 15, 16, 18, 19, 21, 23, 25, 27
Step 2: Count values: 9 (odd number)
Step 3: Find middle position: (9+1) ÷ 2 = 5th position
Step 4: Identify 5th value: 19
Answer: The median is 19
Find the mode of: 7, 8, 9, 7, 6, 8, 7, 5, 8, 7
Answer
Step 1: Count frequency of each value:
- 5 appears 1 time
- 6 appears 1 time
- 7 appears 4 times
- 8 appears 3 times
- 9 appears 1 time
Step 2: Identify most frequent value: 7 appears most (4 times)
Answer: The mode is 7
Calculate the range of: 45, 52, 38, 61, 47, 55, 42
Answer
Step 1: Identify highest value: 61
Step 2: Identify lowest value: 38
Step 3: Apply range formula: Range = Highest - Lowest
Step 4: Calculate: Range = 61 - 38 = 23
Answer: The range is 23
A fair coin is tossed 3 times. What is the probability of getting exactly 2 heads?
Answer
Step 1: List sample space: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT (8 outcomes)
Step 2: Identify favorable outcomes (exactly 2 heads): HHT, HTH, THH (3 outcomes)
Step 3: Apply probability formula: P(E) = Favorable outcomes / Total outcomes
Step 4: Calculate: P(2 heads) = 3/8
Answer: The probability is 3/8 or 0.375
How many ways can 5 students be arranged in a row for a photo?
Answer
Step 1: Identify this as a permutation problem (order matters)
Step 2: Apply factorial formula: n! = n × (n-1) × (n-2) × ... × 1
Step 3: Calculate: 5! = 5 × 4 × 3 × 2 × 1
Step 4: Compute: 5! = 120
Answer: There are 120 different arrangements
In how many ways can you choose 3 books from 8 books on a shelf?
Answer
Step 1: Identify as combination (order doesn't matter)
Step 2: Apply combination formula: C(n,r) = n! / [r!(n-r)!]
Step 3: Substitute: C(8,3) = 8! / [3!(8-3)!] = 8! / (3! × 5!)
Step 4: Simplify: = (8 × 7 × 6) / (3 × 2 × 1) = 336/6 = 56
Answer: There are 56 ways to choose 3 books from 8
A card is drawn from a standard deck. What is the probability of getting a red card or a face card?
Answer
Step 1: Identify events: A = red card, B = face card
Step 2: Calculate P(A): 26 red cards out of 52 = 26/52
Step 3: Calculate P(B): 12 face cards out of 52 = 12/52
Step 4: Calculate P(A and B): 6 red face cards = 6/52
Step 5: Apply addition rule: P(A or B) = P(A) + P(B) - P(A and B)
Step 6: Calculate: = 26/52 + 12/52 - 6/52 = 32/52 = 8/13
Answer: The probability is 8/13
Find the variance of the data set: 2, 4, 6, 8, 10
Answer
Step 1: Find the mean: (2+4+6+8+10)/5 = 30/5 = 6
Step 2: Calculate deviations from mean: (2-6)=-4, (4-6)=-2, (6-6)=0, (8-6)=2, (10-6)=4
Step 3: Square the deviations: 16, 4, 0, 4, 16
Step 4: Find average of squared deviations: (16+4+0+4+16)/5 = 40/5 = 8
Answer: The variance is 8
Two dice are rolled. What is the probability that the sum is 7?
Answer
Step 1: Total possible outcomes: 6 × 6 = 36
Step 2: Find favorable outcomes (sum = 7):
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes
Step 3: Apply probability formula: P(sum=7) = 6/36 = 1/6
Answer: The probability is 1/6 ≈ 0.167
When do you use the combination formula instead of permutation formula?
Answer
Use COMBINATION when:
- Order doesn't matter
- You're selecting/choosing items
- Example: Choosing team members, selecting cards
Use PERMUTATION when:
- Order matters
- You're arranging items
- Example: Seating arrangements, ranking contestants
Key difference: C(n,r) = P(n,r) ÷ r! because combinations eliminate the ordering factor
Calculate P(8,3) - the number of 3-digit arrangements from digits 1,2,3,4,5,6,7,8
Answer
Step 1: Identify as permutation (order matters for arrangements)
Step 2: Apply formula: P(n,r) = n!/(n-r)!
Step 3: Substitute: P(8,3) = 8!/(8-3)! = 8!/5!
Step 4: Simplify: = 8 × 7 × 6 × 5!/5! = 8 × 7 × 6
Step 5: Calculate: = 336
Answer: There are 336 different 3-digit arrangements
Find the median of: 12, 15, 18, 20, 22, 25 (even number of values)
Answer
Step 1: Verify data is arranged: 12, 15, 18, 20, 22, 25 ✓
Step 2: Count values: 6 (even number)
Step 3: Find middle positions: 3rd and 4th values
Step 4: Identify middle values: 18 and 20
Step 5: Calculate median: (18 + 20)/2 = 38/2 = 19
Answer: The median is 19
A bag contains 5 red balls, 3 blue balls, and 2 green balls. What is the probability of drawing a blue ball?
Answer
Step 1: Count total balls: 5 + 3 + 2 = 10 balls
Step 2: Count favorable outcomes (blue balls): 3
Step 3: Apply probability formula: P(blue) = Favorable/Total
Step 4: Calculate: P(blue) = 3/10 = 0.3
Answer: The probability is 3/10 or 0.3 or 30%
Calculate the standard deviation if the variance is 25
Answer
Step 1: Recall relationship: Standard deviation = √(variance)
Step 2: Substitute given variance: SD = √25
Step 3: Calculate: SD = 5
Answer: The standard deviation is 5
Note: Standard deviation is always the positive square root of variance
How many 4-letter words can be formed from MATHEMATICS if no letter is repeated?
Answer
Step 1: Count distinct letters in MATHEMATICS: M-A-T-H-E-M-A-T-I-C-S
Distinct letters: M, A, T, H, E, I, C, S = 8 letters
Step 2: Need 4-letter arrangements from 8 distinct letters
Step 3: Apply permutation: P(8,4) = 8!/(8-4)! = 8!/4!
Step 4: Calculate: = 8 × 7 × 6 × 5 = 1,680
Answer: 1,680 different 4-letter words can be formed
Find the probability of getting at least one head when flipping a coin 3 times
Answer
Method 1 (Complement):
Step 1: P(at least 1 head) = 1 - P(no heads) = 1 - P(all tails)
Step 2: P(all tails) = P(T) × P(T) × P(T) = (1/2)³ = 1/8
Step 3: P(at least 1 head) = 1 - 1/8 = 7/8
Method 2 (Direct): Count favorable outcomes from sample space
Total: 8 outcomes, Favorable: 7 outcomes (all except TTT)
Answer: The probability is 7/8
A committee of 4 people is to be formed from 6 men and 4 women. How many committees can have exactly 2 men?
Answer
Step 1: Need exactly 2 men and 2 women
Step 2: Choose 2 men from 6: C(6,2) = 6!/(2!4!) = 15
Step 3: Choose 2 women from 4: C(4,2) = 4!/(2!2!) = 6
Step 4: Apply multiplication principle: Total = C(6,2) × C(4,2)
Step 5: Calculate: = 15 × 6 = 90
Answer: 90 committees can be formed with exactly 2 men
The mean of 5 numbers is 24. If four numbers are 20, 22, 26, 28, find the fifth number.
Answer
Step 1: Use mean formula: Mean = Sum/Count
Step 2: Calculate total sum: Sum = Mean × Count = 24 × 5 = 120
Step 3: Find sum of known numbers: 20 + 22 + 26 + 28 = 96
Step 4: Find fifth number: Fifth number = Total sum - Sum of known
Step 5: Calculate: Fifth number = 120 - 96 = 24
Answer: The fifth number is 24
Two events A and B are independent. If P(A) = 0.4 and P(B) = 0.3, find P(A and B).
Answer
Step 1: Recall independence rule: For independent events, P(A and B) = P(A) × P(B)
Step 2: Substitute given values: P(A and B) = 0.4 × 0.3
Step 3: Calculate: P(A and B) = 0.12
Answer: P(A and B) = 0.12 or 12%
Note: Independence means one event doesn't affect the other's probability
Diagrams
Statistics and Probability Concept Overview
A comprehensive mind map showing the relationship between all major concepts in Statistics & Probability, helping students understand how different topics connect and build upon each other.
Problem-Solving Decision Tree for Statistics & Probability
A decision tree flowchart that guides students through the process of identifying which formulas and methods to use when solving different types of statistics and probability problems.