Introduction

What is Probability?

Probability is the measure of how likely an event is to occur. It is a fundamental concept in mathematics used to predict the likelihood of different outcomes. In everyday life, we encounter probability when we talk about the chance of rain, the likelihood of rolling a particular number on a die, or the odds of winning a game.

Basic Probability Concepts

  • Outcome: A possible result of a probability experiment (e.g., rolling a 4 on a die).
  • Event: A set of outcomes (e.g., rolling an even number on a die).
  • Probability Scale: Probability is expressed as a number between 0 and 1, where:
  • $$P(\text{Impossible event}) = 0$$
  • $$P(\text{Certain event}) = 1$$
  • Values between 0 and 1 indicate varying degrees of likelihood. Probability is calculated as:
    $$
    P(\text{Event}) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}
    $$

Why Learn Probability?

Understanding probability helps students:

  • Make informed decisions based on the likelihood of events.
  • Understand randomness and chance in real-world situations.
  • Interpret and analyse data involving probabilities.

Question Sets

Easy Level Questions

Q1-Q10: Basic Probability

  1. What is the probability of flipping a coin and getting heads?
  2. If you roll a 6-sided die, what is the probability of rolling a 3?
  3. What is the probability of choosing a red marble from a bag containing 5 red and 5 blue marbles?
  4. What is the probability of rolling an even number on a 6-sided die?
  5. If you pick a card randomly from a standard deck of 52 playing cards, what is the probability of drawing a heart?
  6. What is the probability of choosing a vowel from the letters in the word “MATHS”?
  7. You flip a coin twice. What is the probability of getting two tails?
  8. A bag contains 4 red, 3 blue, and 5 yellow balls. What is the probability of picking a red ball?
  9. What is the probability of rolling a number greater than 4 on a 6-sided die?
  10. If you spin a spinner with 4 equal sections labelled 1, 2, 3, and 4, what is the probability of landing on a 1?

Q11-Q20: Slightly Complex Scenarios

  1. A bag contains 5 green, 2 yellow, and 3 red marbles. What is the probability of picking a yellow marble?
  2. What is the probability of rolling a number less than 3 on a 6-sided die?
  3. You flip a coin three times. What is the probability of getting exactly one head?
  4. A deck of cards has 52 cards. What is the probability of drawing a king?
  5. In a classroom of 20 students, 12 are girls. What is the probability of picking a girl at random?
  6. A jar contains 10 chocolates, 4 of which are dark chocolates. What is the probability of picking a dark chocolate?
  7. You randomly select a letter from the word “PROBABILITY.” What is the probability of selecting a “B”?
  8. A spinner has 8 equal sections, 4 of which are blue. What is the probability of landing on blue?
  9. What is the probability of drawing an ace or a king from a standard deck of 52 cards?
  10. What is the probability of not rolling a 5 on a 6-sided die?

Medium Level Questions

Q1-Q10: Intermediate Scenarios

  1. A bag contains 4 green, 3 yellow, 5 red marbles. What is the probability of not picking a green marble?
  2. A spinner has 10 equal sections, numbered 1 to 10. What is the probability of landing on a prime number?
  3. What is the probability of rolling an odd number or a 4 on a 6-sided die?
  4. You roll two dice. What is the probability that the sum is 7?
  5. In a group of 30 students, 12 like football, 10 like basketball, and 8 like both sports. What is the probability of selecting a student who likes only football?
  6. A bag contains 6 red, 4 blue, and 2 green balls. What is the probability of picking a blue or green ball?
  7. What is the probability of drawing a red card or a face card from a standard deck of 52 cards?
  8. What is the probability of getting at least one head when flipping two coins?
  9. You pick two cards from a standard deck of 52 cards without replacement. What is the probability that both are aces?
  10. A bag contains 5 black and 3 white marbles. What is the probability of picking a black marble, replacing it, and then picking a white marble?

Q11-Q20: Intermediate Problem Solving

  1. What is the probability of rolling a sum of 10 with two dice?
  2. A jar contains 5 red, 4 yellow, and 6 green marbles. What is the probability of not picking a yellow marble?
  3. You roll two dice. What is the probability of rolling doubles?
  4. A bag contains 3 red, 5 blue, and 7 yellow balls. What is the probability of picking a red or yellow ball?
  5. What is the probability of drawing a spade or a queen from a deck of cards?
  6. A spinner has 8 equal sections labelled with numbers. What is the probability of landing on an even number or a number greater than 6?
  7. What is the probability of drawing two hearts from a standard deck of cards without replacement?
  8. What is the probability of rolling a 3 or 6 on two dice?
  9. A bag contains 10 sweets: 3 orange, 4 strawberry, and 3 lemon. What is the probability of picking a strawberry or lemon sweet?
  10. You flip three coins. What is the probability of getting exactly two heads?

Hard Level Questions

Q1-Q10: Advanced Probability Concepts

  1. You roll two dice. What is the probability of the product of the two numbers being even?
  2. You draw two cards from a standard deck without replacement. What is the probability that one is a spade and the other is a heart?
  3. A bag contains 4 blue, 5 red, and 6 yellow balls. You pick two balls without replacement. What is the probability that both balls are yellow?
  4. What is the probability of rolling a sum of 9 on two dice?
  5. What is the probability of drawing a jack, queen, or king from a standard deck of 52 cards?
  6. You pick two cards from a deck of 52 without replacement. What is the probability that both cards are face cards?
  7. What is the probability of drawing an ace or a spade from a standard deck of 52 cards?
  8. If a bag contains 5 red, 3 green, and 2 blue balls, what is the probability of drawing two red balls consecutively without replacement?
  9. You roll two dice. What is the probability of rolling a sum of 5 or a sum of 9?
  10. What is the probability of picking three red marbles in a row from a bag containing 6 red and 4 blue marbles, without replacement?

Q11-Q20: Complex Problem Solving

  1. What is the probability of getting at least one 6 when rolling two dice?
  2. A bag contains 7 green and 5 red marbles. Two marbles are drawn without replacement. What is the probability that the first is green and the second is red?
  3. You flip four coins. What is the probability of getting exactly three heads?
  4. What is the probability of drawing two cards from a deck of 52 cards and having one be a diamond and the other a heart?
  5. You roll two dice. What is the probability that the sum is either 7 or 11?
  6. In a class of 20 students, 12 are boys. What is the probability of randomly selecting two boys in a row without replacement?
  7. You have a deck of 52 cards. What is the probability of drawing two consecutive kings without replacement?
  8. What is the probability of rolling a prime number on two dice?
  9. You pick three cards from a deck without replacement. What is the probability of drawing exactly one king?
  10. What is the probability of getting at least one tail when flipping three coins?

Answers and Explanations

Easy Level

Q1-Q10: Basic Probability

  1. Answer:
    $$P(\text{Heads}) = \frac{1}{2}$$
    Explanation: There are 2 outcomes (heads or tails), and heads is 1 of those outcomes.
  2. Answer:
    $$P(\text{Rolling a 3}) = \frac{1}{6}$$
    Explanation: There are 6 possible outcomes when rolling a die, and 3 is one of them.
  3. Answer:
    $$P(\text{Red}) = \frac{5}{10} = \frac{1}{2}$$
    Explanation: There are 5 red marbles out of a total of 10 marbles (5 red and 5 blue).
  4. Answer:
    $$P(\text{Even number}) = \frac{3}{6} = \frac{1}{2}$$
    Explanation: The even numbers on a 6-sided die are 2, 4, and 6 (3 outcomes).
  5. Answer:
    $$P(\text{Heart}) = \frac{13}{52} = \frac{1}{4}$$
    Explanation: There are 13 hearts in a deck of 52 cards.
  6. Answer:
    $$P(\text{Vowel}) = \frac{1}{5}$$
    Explanation: There is 1 vowel (A) in the word “MATHS,” and the total number of letters is 5.
  7. Answer:
    $$P(\text{Two tails}) = \frac{1}{4}$$
    Explanation: The possible outcomes when flipping a coin twice are HH, HT, TH, and TT. Only TT is two tails.
  8. Answer:
    $$P(\text{Red}) = \frac{4}{12} = \frac{1}{3}$$
    Explanation: There are 4 red balls out of a total of 12 balls (4 red, 3 blue, and 5 yellow).
  9. Answer:
    $$P(\text{> 4}) = \frac{2}{6} = \frac{1}{3}$$
    Explanation: The numbers greater than 4 on a die are 5 and 6 (2 outcomes).
  10. Answer:
    $$P(\text{Landing on 1}) = \frac{1}{4}$$
    Explanation: There is 1 section labelled 1 out of 4 equal sections on the spinner.

Q11-Q20: Slightly Complex Scenarios

  1. Answer:
    $$P(\text{Yellow}) = \frac{2}{14} = \frac{1}{7}$$
    Explanation: There are 2 yellow marbles out of a total of 14 marbles (5 green, 2 yellow, and 7 red).
  2. Answer:
    $$P(\text{< 3}) = \frac{2}{6} = \frac{1}{3}$$
    Explanation: The numbers less than 3 on a die are 1 and 2 (2 outcomes).
  3. Answer:
    $$P(\text{Exactly one head}) = \frac{3}{8}$$
    Explanation: The possible outcomes when flipping three coins are HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. The outcomes with exactly one head are HTT, THT, and TTH.
  4. Answer:
    $$P(\text{King}) = \frac{4}{52} = \frac{1}{13}$$
    Explanation: There are 4 kings in a deck of 52 cards.
  5. Answer:
    $$P(\text{Girl}) = \frac{12}{20} = \frac{3}{5}$$
    Explanation: There are 12 girls out of a total of 20 students.
  6. Answer:
    $$P(\text{Dark chocolate}) = \frac{4}{10} = \frac{2}{5}$$
    Explanation: There are 4 dark chocolates out of a total of 10 chocolates.
  7. Answer:
    $$P(\text{B}) = \frac{2}{11}$$
    Explanation: There are 2 B’s in the word “PROBABILITY,” which has 11 letters total.
  8. Answer:
    $$P(\text{Blue}) = \frac{4}{8} = \frac{1}{2}$$
    Explanation: There are 4 blue sections out of a total of 8 sections on the spinner.
  9. Answer:
    $$P(\text{Ace or King}) = \frac{4 + 4}{52} = \frac{8}{52} = \frac{2}{13}$$
    Explanation: There are 4 aces and 4 kings in a deck of 52 cards.
  10. Answer:
    $$P(\text{Not 5}) = 1 – P(\text{5}) = 1 – \frac{1}{6} = \frac{5}{6}$$
    Explanation: The probability of rolling a 5 is $\frac{1}{6}$, so the probability of not rolling a 5 is $\frac{5}{6}$.

Medium Level

Q1-Q10: Intermediate Scenarios

  1. Answer:
    $$P(\text{Not green}) = 1 – P(\text{Green}) = 1 – \frac{4}{12} = \frac{8}{12} = \frac{2}{3}$$
    Explanation: There are 4 green marbles out of 12 total marbles.
  2. Answer:
    $$P(\text{Prime number}) = \frac{4}{10} = \frac{2}{5}$$
    Explanation: The prime numbers between 1 and 10 are 2, 3, 5, and 7.
  3. Answer:
    $$P(\text{Odd or 4}) = P(\text{Odd}) + P(4) = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$$
    Explanation: The odd numbers are 1, 3, and 5.
  4. Answer:
    $$P(\text{Sum of 7}) = \frac{6}{36} = \frac{1}{6}$$
    Explanation: The combinations for a sum of 7 are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).
  5. Answer:
    $$P(\text{Only football}) = \frac{12 – 8}{30} = \frac{4}{30} = \frac{2}{15}$$
    Explanation: There are 4 students who like only football.
  6. Answer:
    $$P(\text{Blue or Green}) = \frac{4 + 2}{12} = \frac{6}{12} = \frac{1}{2}$$
    Explanation: There are 4 blue and 2 green balls out of a total of 12 balls.
  7. Answer:
    $$P(\text{Red or Face}) = \frac{26 + 12 – 3}{52} = \frac{35}{52}$$
    Explanation: There are 26 red cards and 12 face cards, but 3 are both.
  8. Answer:
    $$P(\text{At least one head}) = 1 – P(\text{No heads}) = 1 – \frac{1}{4} = \frac{3}{4}$$
    Explanation: The only outcome with no heads is Tails, Tails.
  9. Answer:
    $$P(\text{Both aces}) = \frac{4}{52} \times \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$$
    Explanation: The first ace has 4 options, and the second has 3 left in a total of 51 cards.
  10. Answer:
    $$P(\text{Black, then White}) = \frac{5}{8} \times \frac{3}{8} = \frac{15}{64}$$
    Explanation: The first pick is a black ball and the second pick is a white ball.

Q11-Q20: Intermediate Problem Solving

  1. Answer:
    $$P(\text{Sum of 10}) = \frac{3}{36} = \frac{1}{12}$$
    Explanation: The combinations are (4,6), (5,5), and (6,4).
  2. Answer:
    $$P(\text{Not yellow}) = 1 – P(\text{Yellow}) = 1 – \frac{4}{15} = \frac{11}{15}$$
    Explanation: There are 4 yellow marbles in a total of 15 marbles.
  3. Answer:
    $$P(\text{Doubles}) = \frac{6}{36