Simple Probability Problems MCQs Quiz | Class 10

This quiz is designed for Class X students studying Mathematics (Code 041), specifically Unit VII: Statistics & Probability. The topic covered is Simple Probability Problems MCQs Quiz, focusing on coin, dice, and card-type single event problems. Test your understanding by attempting all 10 multiple-choice questions. Once completed, you can submit your answers to view your score and download a detailed PDF of questions with correct answers.

Understanding Simple Probability for Class 10

Probability is a branch of mathematics that deals with the likelihood of an event occurring. In Class 10, you are introduced to simple probability problems, which involve calculating the chances of single events happening in various scenarios like tossing coins, rolling dice, or drawing cards.

Key Concepts of Probability

  • Experiment: An action or process that leads to well-defined outcomes (e.g., tossing a coin, rolling a die).
  • Outcome: A possible result of an experiment (e.g., Head when tossing a coin, 4 when rolling a die).
  • Sample Space (S): The set of all possible outcomes of an experiment (e.g., {H, T} for a coin toss, {1, 2, 3, 4, 5, 6} for a die roll).
  • Event (E): A subset of the sample space; a collection of some outcomes (e.g., getting an even number when rolling a die: {2, 4, 6}).
  • Favorable Outcomes: The outcomes that satisfy the conditions of a particular event.

The Probability Formula

The probability of an event E, denoted as P(E), is calculated using the formula:

P(E) = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)

Important Properties of Probability

  • The probability of an event always lies between 0 and 1, inclusive: 0 ≤ P(E) ≤ 1.
  • P(E) = 0 for an impossible event (an event that cannot happen).
  • P(E) = 1 for a sure or certain event (an event that is certain to happen).
  • The sum of probabilities of all elementary events of an experiment is 1.
  • For any event E, P(E) + P(not E) = 1, where P(not E) is the probability of the event not happening (also called the complementary event).

Common Types of Probability Problems

1. Coin-Based Problems

When a fair coin is tossed, there are two equally likely outcomes: Head (H) and Tail (T).

  • One Coin Toss: Sample space = {H, T}. Total outcomes = 2.
  • Two Coin Tosses: Sample space = {HH, HT, TH, TT}. Total outcomes = 4.
  • Three Coin Tosses: Sample space = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}. Total outcomes = 8.

Example: Probability of getting exactly one head in two coin tosses: Outcomes are {HT, TH}. Favorable outcomes = 2. P(exactly one head) = 2/4 = 1/2.

2. Dice-Based Problems

When a fair die is rolled, there are six equally likely outcomes, the numbers 1, 2, 3, 4, 5, 6.

  • One Die Roll: Sample space = {1, 2, 3, 4, 5, 6}. Total outcomes = 6.
  • Two Dice Rolls: Total outcomes = 6 x 6 = 36.

Example: Probability of getting an even number on a single die roll: Outcomes are {2, 4, 6}. Favorable outcomes = 3. P(even number) = 3/6 = 1/2.

3. Card-Based Problems

A standard deck of playing cards has 52 cards, divided into four suits:

  • Spades (♠): 13 cards (Black)
  • Clubs (♣): 13 cards (Black)
  • Hearts (♥): 13 cards (Red)
  • Diamonds (♦): 13 cards (Red)

Each suit contains an Ace (A), King (K), Queen (Q), Jack (J), and number cards from 2 to 10.

  • Face Cards: King, Queen, Jack (3 in each suit, so 3 x 4 = 12 face cards in total).
  • Aces: 4 aces (one in each suit).
  • Red Cards: 26 (Hearts + Diamonds).
  • Black Cards: 26 (Spades + Clubs).

Example: Probability of drawing a King from a well-shuffled deck: There are 4 Kings. P(King) = 4/52 = 1/13.

Quick Revision Points

  • P(Event) = (Favorable Outcomes) / (Total Outcomes)
  • Probability is always between 0 and 1.
  • P(E) + P(not E) = 1.
  • For coin tosses, total outcomes are 2^n (n is number of coins).
  • For die rolls, total outcomes are 6^n (n is number of dice).
  • Remember the composition of a 52-card deck (suits, colors, face cards).

Practice Questions (Self-Assessment)

  1. A letter is chosen at random from the word “MATHEMATICS”. What is the probability that it is a vowel?
  2. A bag contains 6 red, 4 blue, and 2 yellow balls. A ball is drawn at random. What is the probability that the ball drawn is not red?
  3. In a leap year, what is the probability of having 53 Sundays?
  4. Two dice are thrown simultaneously. What is the probability that the sum of the numbers appearing on the dice is 7?
  5. From a well-shuffled deck of 52 playing cards, a card is drawn at random. Find the probability of getting a black Queen.

By mastering these simple concepts and practicing various problems, you can confidently solve probability questions in your examinations.