USTET Mathematics — Statistics & ProbabilityFlash Cards

Find the mean of the data set: 85, 92, 78, 96, 88, 91, 84

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

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

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

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?

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?

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?

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?

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

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?

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?

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

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)

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?

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

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?

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

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?

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.

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).

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