Probability Class 10 Chapter Notes (CBSE Maths Chapter 15)

Key Terms in Probability

Experiment

An action or operation which can produce some well-defined outcomes. Example: Tossing a coin.

Sample Space (S)

The set of all possible outcomes of an experiment. For a die roll, S = {1, 2, 3, 4, 5, 6}.

Event (E)

A single outcome or a set of outcomes of an experiment. Example: Getting an even number when rolling a die, E = {2, 4, 6}.

Elementary Event

An event having only one outcome. Example: Getting a '4' on a die roll. The sum of probabilities of all elementary events of an experiment is 1.

Sure Event (or Certain Event)

An event that is certain to occur. Its probability is 1. Example: Getting a number less than 7 when rolling a standard die.

Impossible Event

An event that cannot occur. Its probability is 0. Example: Getting a '7' when rolling a standard die.

Complementary Event (not E or E')

The event representing the non-occurrence of event E. The probability of a complementary event is P(not E) = 1 - P(E).

Key Formulas & Must-Remember Points

• {"point":"The theoretical probability of an event E is given by the formula:","formula":"P(E) = (Number of outcomes favourable to E) / (Total number of possible outcomes)"}
• {"point":"The probability of any event E always lies between 0 and 1 (inclusive).","formula":"0 ≤ P(E) ≤ 1"}
• {"point":"The probability of a sure event is 1."}
• {"point":"The probability of an impossible event is 0."}
• {"point":"For any event E, the sum of its probability and the probability of its complement is 1.","formula":"P(E) + P(not E) = 1"}
• {"point":"A standard deck of playing cards has 52 cards, divided into 4 suits (Spades ♠, Hearts ♥, Diamonds ♦, Clubs ♣)."}
• {"point":"There are 26 red cards (Hearts, Diamonds) and 26 black cards (Spades, Clubs) in a deck."}
• {"point":"Each suit has 13 cards: A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J (Jack), Q (Queen), K (King)."}
• {"point":"There are 12 face cards in a deck (3 per suit: J, Q, K)."}
• {"point":"When two dice are rolled, the total number of outcomes is 6 x 6 = 36."}

Understanding Sample Space, Outcomes, and Events

In probability, every problem starts with an experiment, which is any process with a result that is uncertain. For example, tossing a coin is an experiment. The possible results of an experiment are called outcomes. When you toss a coin, the two possible outcomes are 'Heads' (H) or 'Tails' (T).

The collection of ALL possible outcomes is known as the sample space, usually denoted by 'S'. For a single coin toss, the sample space is S = {H, T}. For a single roll of a standard six-sided die, the sample space is S = {1, 2, 3, 4, 5, 6}.

An event is a subset of the sample space; it's the specific outcome or group of outcomes you are interested in. If the experiment is rolling a die, an event (let's call it E) could be 'getting an odd number'. The outcomes favourable to this event are {1, 3, 5}. To find the probability of this event, P(E), you use the main formula: divide the number of favourable outcomes (3) by the total number of outcomes in the sample space (6). So, P(E) = 3/6 = 1/2. An event with just one outcome, like 'getting a 4', is called an elementary event.

Sure, Impossible, and Complementary Events

Worked Examples

• {"title":"Example 1: Tossing Two Coins","bodyMarkdown":"Problem: Two coins are tossed simultaneously. What is the probability of getting at least one head?\n\nSolution:\n1. Sample Space (S): {HH, HT, TH, TT}. Total outcomes = 4.\n2. Favourable Outcomes (E): 'At least one head' means one head or two heads. So, E = {HH, HT, TH}. Number of favourable outcomes = 3.\n3. Probability P(E):\n P(E) = (Number of favourable outcomes) / (Total outcomes) = 3/4."}
• {"title":"Example 2: Drawing a Card","bodyMarkdown":"Problem: A card is drawn from a well-shuffled deck of 52 cards. What is the probability that the card is a black face card?\n\nSolution:\n1. Total Outcomes: 52 cards.\n2. Favourable Outcomes (E): There are 12 face cards in total (J, Q, K of 4 suits). Half of these are black (Spades and Clubs). So, the black face cards are Jack of Spades, Queen of Spades, King of Spades, Jack of Clubs, Queen of Clubs, King of Clubs. Number of favourable outcomes = 6.\n3. Probability P(E):\n P(E) = 6/52 = 3/26."}

Exam and Scoring Tips

Probability questions are generally high-scoring but prone to silly mistakes.

• Always write the formula: P(E) = (Favourable Outcomes) / (Total Outcomes). This can fetch you marks even if the final calculation is wrong.
• Card Problems: Memorize the structure of a 52-card deck. A common mistake is miscounting face cards (12) or confusing them with Aces. Remember, Aces are not face cards.
• 'At Least' vs 'At Most': Read the question carefully. 'At least one' means one or more. 'At most one' means one or zero.
• Final Answer: Always simplify your final fraction to its lowest terms (e.g., write 1/2 instead of 3/6). Your answer must be between 0 and 1.

Practice Questions with Solutions

• What is the probability of an event that is certain to happen? 1.
• A bag contains 5 red balls and 3 blue balls. What is the probability of drawing a blue ball? Total balls = 5 + 3 = 8. Favourable outcomes (blue) = 3. So, P(blue) = 3/8.
• If the probability of winning a game is 0.6, what is the probability of losing it? P(losing) = 1 - P(winning) = 1 - 0.6 = 0.4.
• A die is thrown once. What is the probability of getting a prime number? Sample space = {1, 2, 3, 4, 5, 6}. Prime numbers = {2, 3, 5}. Favourable outcomes = 3. Total outcomes = 6. P(prime) = 3/6 = 1/2.

Frequently Asked Questions