MP Board · Class 10 · Mathematics · ProbabilityA box contains 90 discs which are numbered from 1 to 90. If one disc is drawn at random from the box, find the probability that it bears: (i) a two-digit number, (ii) a perfect square number, (iii) a number divisible by 5. Also, explain the theoretical concept of empirical probability used in solving such random experiments.

Numerical Solution:

Total number of possible outcomes:\nSince there are 90 discs numbered from 1 to 90, the total number of elementary events in the sample space is $n(S) = 90$.

(i) Probability of getting a two-digit number:

• The two-digit numbers from 1 to 90 are 10, 11, 12, ..., 90.
• Total number of two-digit numbers = $90 - 9 = 81$.
• Let $E_1$ be the event of getting a two-digit number. Therefore, $n(E_1) = 81$.
• $\text{Probability } P(E_1) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{n(E_1)}{n(S)} = \frac{81}{90} = \frac{9}{10}$.

(ii) Probability of getting a perfect square number:

• The perfect square numbers between 1 and 90 are 1, 4, 9, 16, 25, 36, 49, 64, and 81.
• Total number of perfect square numbers = 9.
• Let $E_2$ be the event of getting a perfect square number. Therefore, $n(E_2) = 9$.
• $\text{Probability } P(E_2) = \frac{n(E_2)}{n(S)} = \frac{9}{90} = \frac{1}{10}$.

(iii) Probability of getting a number divisible by 5:

• The numbers between 1 and 90 that are divisible by 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, and 90.
• Total number of such numbers = 18.
• Let $E_3$ be the event of getting a number divisible by 5. Therefore, $n(E_3) = 18$.
• $\text{Probability } P(E_3) = \frac{n(E_3)}{n(S)} = \frac{18}{90} = \frac{1}{5}$.

Theoretical Concept of Probability:

• Definition of Probability: Probability is a quantitative measure of the likelihood that a particular event will occur. It is expressed as a number between 0 and 1, inclusive.
• Equally Likely Outcomes: Outcomes of a sample space are called equally likely if each outcome has the same chance of occurrence. For example, drawing any disc from the box has the exact same physical chance, making them equally likely.
• Formula for Theoretical Probability: The theoretical probability (also called classical probability) of an event $E$, denoted by $P(E)$, is defined as: $$P(E) = \frac{\text{Number of outcomes favorable to } E}{\text{Total number of all possible outcomes of the experiment}}$$
• Complementary Events: The complement of an event $E$, denoted by $\bar{E}$ or $E'$, represents the event 'not $E$'. The sum of the probabilities of an event and its complementary event is always equal to 1, i.e., $P(E) + P(\bar{E}) = 1$.
• Range of Probability: The probability of any event $E$ always satisfies the inequality $0 \le P(E) \le 1$. An event that is certain to happen has a probability of 1, and an event that is impossible has a probability of 0.