CBSE Class 10 Maths Chapter 15: Probability Exercise 15.1

Theoretical Probability Concepts

Theoretical probability of an event E, denoted as P(E), is defined as the ratio of the number of outcomes favorable to E to the total number of possible outcomes in the sample space, assuming all outcomes are equally likely. Mathematically:

P(E) = (Number of favorable outcomes) / (Total number of possible outcomes)

In Exercise 15.1, we study several key axioms:

1. Range of Probability: The probability of any event is a real number ranging from 0 to 1, represented as 0 <= P(E) <= 1.
2. Sure and Impossible Events: An event guaranteed to occur has a probability of 1 (Sure Event). An event that can never occur has a probability of 0 (Impossible Event).
3. Complementary Events: For any event E, the event 'not E' (denoted as E') is its complement. The sum of their probabilities is always 1: P(E) + P(E') = 1.

Step-by-Step Problem Solving Strategy

1. Identify the Sample Space — Count the total number of possible outcomes for the given random experiment (e.g., throwing a die, drawing a card, tossing coins).
2. Determine Favorable Outcomes — Identify and count all outcomes that satisfy the specific condition defined by event E.
3. Apply the Classical Probability Formula — Divide the number of favorable outcomes by the total number of outcomes. Simplify the fraction to its lowest terms.
4. Calculate Complementary Events — If asked to find the probability of 'not E', simply calculate 1 - P(E) instead of counting all non-favorable outcomes manually.

Common Mistakes and Board Exam Tips

• Probability Range Limits: Keep in mind that a probability value can never be negative (e.g., -0.5) or greater than 1 (e.g., 1.5 or 120%). If your calculation yields such results, re-evaluate your steps.
• Prime Numbers on a Die: Students often mistakenly count '1' as a prime number. The prime numbers on a standard six-sided die are 2, 3, and 5.
• Simplifying Fractions: Always reduce your final fractional answer to its lowest terms to avoid minor marks deductions.

Practice Questions with Solutions

• Q: If P(E) = 0.05, what is the probability of 'not E'? A: Step 1: Use the complementary event formula: P(E) + P(not E) = 1. Step 2: Substitute the given value: 0.05 + P(not E) = 1. Step 3: Solve for P(not E): P(not E) = 1 - 0.05 = 0.95. Final answer: The probability of 'not E' is 0.95.
• Q: A bag contains 3 red balls and 5 black balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is (i) red, (ii) not red? A: Step 1: Calculate the total number of outcomes. Total balls = 3 (red) + 5 (black) = 8. Step 2: For (i) Red, favorable outcomes = 3. P(Red) = 3/8. Step 3: For (ii) Not red, use the complementary formula: P(not Red) = 1 - P(Red) = 1 - 3/8 = 5/8. Final answer: (i) P(Red) = 3/8, (ii) P(not Red) = 5/8.
• Q: A die is thrown once. Find the probability of getting (i) a prime number, (ii) a number lying between 2 and 6. A: Step 1: Write down the total outcomes of a single die throw: {1, 2, 3, 4, 5, 6}. Total outcomes = 6. Step 2: (i) Prime numbers are {2, 3, 5}. Favorable outcomes = 3. P(Prime) = 3/6 = 1/2. Step 3: (ii) Numbers lying strictly between 2 and 6 are {3, 4, 5}. Favorable outcomes = 3. P(Between 2 and 6) = 3/6 = 1/2. Final answer: (i) P(Prime) = 1/2, (ii) P(Between 2 and 6) = 1/2.
• Q: Which of the following cannot be the probability of an event? (A) 2/3 (B) -1.5 (C) 15% (D) 0.7 A: Step 1: Recall that the value of probability P(E) must always satisfy 0 <= P(E) <= 1. Step 2: Check option (A): 2/3 ≈ 0.67 (valid). Step 3: Check option (B): -1.5 is negative, which is strictly less than 0 (invalid). Step 4: Check option (C): 15% = 0.15 (valid). Step 5: Check option (D): 0.7 (valid). Final answer: (B) -1.5 cannot be the probability of an event.

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