Hard Level Worksheet

Very Short Answer Questions (1 Mark Each)

(1) A bag contains 10 identical balls marked 1 to 10. One ball is drawn. What is the probability that the number is a prime?

Probability =

Correct! 4 prime numbers out of 10 total numbers.

(2) A card is drawn at random from a pack of 52 cards. Find the probability that it is neither an ace nor a king.

P =

Perfect! Excluded 4 aces and 4 kings.

(3) A die is rolled. What is the probability of getting an odd prime number?

Probability =

Excellent! Note that 2 is prime but even.

(4) A coin is tossed thrice. What is the probability of getting exactly two heads?

P =

Great! 3 ways out of 23 = 8 total outcomes.

(5) Two dice are thrown. What is the probability of getting numbers whose product is even?

P(product even) =

Excellent! Used complement rule effectively.

Short Answer Questions (2 Marks Each)

Answer each question with clear steps

(1) A bag contains 5 red, 4 green, and 3 blue balls. A ball is drawn at random. Find the probability of getting: (a) Not a green ball (b) Neither red nor blue ball

(a) P(Not green) =

(b) P(Neither red nor blue) =

Perfect! These probabilities sum to 1 as expected.

(2) A die is rolled twice. Find the probability that the numbers obtained have: (a) Sum divisible by 3 (b) Sum greater than 9

(a) P =

(b) P =

Excellent systematic counting!

(3) From a well-shuffled pack of 52 cards, a card is drawn. Find the probability of getting: (a) Neither a red card nor a king (b) A card that is either a spade or a face card

(a) P(Neither red nor king) =

(b) P(Either a spade or a face card) =

Great use of inclusion-exclusion principle!

(4) Two coins are tossed together. Find the probability of getting: (a) At least one tail (b) No head

(a) P(At least one tail) =

(b) P(No head) =

Perfect! Note: P(at least one tail) = 1 - P(no tail).

(5) A box contains 15 bulbs out of which 5 are defective. Two bulbs are chosen at random one after the other without replacement. Find the probability that both are good.

P(Both good) =

Excellent! Second draw has 14 total and 9 good remaining.

Long Answer Questions (4 Marks Each)

Note: Answer each question with complete calculations and clear reasoning.

(1) A bag contains 6 red, 4 white, and 5 black balls. A ball is drawn at random. Find the probability of getting: (a) A red or black ball (b) Neither red nor black ball. If two balls are drawn without replacement, find the probability that both are black.

(a) P(Red or Black) =

(b) P(Neither red nor black) =

P(Both black) =

(2) Two dice are thrown. Find the probability of getting: (a) Sum equal to 11 (b) A doublet of even numbers (c) Sum less than or equal to 4 (d) At least one 6

(a) Sum = 11: P =

(b) Even doublets: P =

(d) At least one 6: P =

(3) A card is drawn from a pack of 52 cards. Find the probability of getting: (a) Neither a black card nor a face card (b) A non-face card (c) A number card of hearts (d) An ace of spades

(a) Neither black nor face: P =

(b) Non-face cards: P =

(d) Ace of spades: P =

(4) A die is thrown thrice. Find the probability of getting: (a) Exactly two sixes (b) At least two sixes (c) No six (d) At most one six

(a) P(Exactly 2 sixes) =

(b) P(At least 2 sixes) =

(d) P(At most 1 six) =

(5) A bag contains 8 red, 6 green, and 4 blue balls. Three balls are drawn one after the other without replacement. Find the probability that: (a) All are red (b) All are green (c) Two are red and one is green (d) At least one is blue

(a) P(All red) =

(b) P(All green) =

(d) P(At least 1B) =

Part B: Objective Questions (1 Mark Each)

Choose the correct answer and write the option (a/b/c/d)

(1) If two dice are thrown, the probability of getting doublet (same number on both) is:

(a) 112 (b) 136 (c) 16 (d) 118

(a) 1/12

(b) 1/36

(d) 1/18

Correct! 6 doublets out of 36 total outcomes = 636 = 16.

(2) A card is drawn from 52 cards. The probability of getting a number card is:

(a) 913 (b) 313 (c) 1013 (d) 1213

(a) 9/13

(b) 3/13

(d) 12/13

Correct! Number cards 2-10 = 36 cards

(3) A coin is tossed 4 times. The probability of getting all heads is:

(a) 18 (b) 116 (c) 132 (d) 14

(a) 1/8

(b) 1/16

(d) 1/4

Correct! 124 = 116 for all heads in 4 tosses.

(4) A bag contains 5 white and 7 black balls. One ball is drawn. The probability of getting a black ball is:

(a) 512 (b) 712 (c) 112 (d) 12

(a) 5/12

(b) 7/12

(d) 1/2

Correct! 7 black balls out of 12 total balls.

(5) A die is thrown. The probability of getting a square number is:

(a) 16 (b) 26 (c) 36 (d) 46

(a) 1/6

(b) 2/6

(d) 4/6

Correct! Square numbers on a die: 1, 4. So 26 = 13.

(6) A card is drawn from 52 cards. Probability of getting neither red nor black card is:

(a) 0 (b) 12 (c) 1 (d) 14

(a) 0

(b) 1/2

(d) 1/4

Correct! All cards are either red or black, so P(neither) = 0.

(7) The probability of an event that cannot happen is:

(a) 0 (b) 1 (c) 12 (d) Between 0 and 1

(a) 0

(b) 1

(d) Between 0 and 1

Correct! Impossible events have probability 0.

(8) A die is rolled. Probability of getting a number greater than 2 is:

(a) 26 (b) 36 (c) 46 (d) 56

(a) 2/6

(b) 3/6

(d) 5/6

Correct! Numbers > 2: {3, 4, 5, 6} = 4 numbers.

(9) A letter is chosen at random from the word "PROBABILITY". The probability of choosing B is:

(a) 111 (b) 211 (c) 311 (d) 411

(a) 1/11

(b) 2/11

(d) 4/11

Correct! PROBABILITY has 2 B's out of 11 letters.

(10) Two dice are thrown. The probability of getting sum equal to 12 is:

(a) 112 (b) 136 (c) 16 (d) 236

(a) 1/12

(b) 1/36

(d) 2/36

Correct! Only (6,6) gives sum 12. So 136.

Fair coin tosses

Cards from same deck

Multiple die rolls

P(A and B) = P(A) × P(B)

P(A or B) = P(A) + P(B) - P(A and B)

Dice outcomes

Sequential ball drawing

Drawing without replacement

Dependent Events

Independent Events

Probability Rules

Advanced Probability Challenge

Determine whether these statements are True or False:

P(A and B) ≤ P(A) for any events A and B

If P(A) = 0.6 and P(B) = 0.5, then P(A or B) = 1.1

For independent events: P(A and B) = P(A) × P(B)

P(A) + P(not A) = 1 for any event A

Drawing cards without replacement gives independent events

The complement rule: P(at least one) = 1 - P(none)

Advanced Probability Quiz

🎉 Congratulations! What You've Mastered:

You have successfully completed the "Probability" hard worksheet and learned:

(1) Advanced Sample Spaces: Analyzing complex scenarios with multiple trials and conditional outcomes

(2) Compound Events: Calculating probabilities involving "and", "or", "at least", "at most" conditions

(3) Conditional Probability: Understanding dependent events and how previous outcomes affect subsequent probabilities

(4) Without Replacement: Solving problems where the sample space changes after each draw

(5) Complement Rule: Using P(A) = 1 - P(not A) for efficient calculation of complex events

(6) Independent vs Dependent: Distinguishing between events that do and don't affect each other

(7) Combinatorial Probability: Applying counting principles to calculate probabilities systematically

(8) Multiple Trials: Analyzing scenarios with repeated experiments like multiple coin tosses or die rolls

(9) Inclusion-Exclusion Principle: Calculating P(A or B) when events may overlap

(10) Prime and Composite Numbers: Applying number theory concepts in probability contexts

(11) Card Problems: Mastering complex scenarios involving multiple conditions and restrictions

(12) Strategic Problem Solving: Choosing optimal approaches (direct calculation vs. complement) based on problem structure

(13) Binomial Probability: Understanding patterns in repeated independent trials

(14) Advanced Counting: Systematically enumerating favorable outcomes in complex sample spaces

(15) Real-world Applications: Connecting probability theory to practical scenarios and decision-making

Outstanding work! You now have mastery over advanced probability concepts and problem-solving techniques!

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