Introduction to Probability Ex 15.1 Class 9 NCERT Solutions

Core Concepts of Empirical Probability

In CBSE Class 9, we study Experimental or Empirical Probability, which relies directly on actual trials and observed data. This differs from classical probability, where we predict outcomes without performing the experiment.

Let $E$ be an event in an experiment. The empirical probability $P(E)$ of the occurrence of event $E$ is given by the simple formula:

$P(E) = \frac{\text{Number of trials in which the event happened}}{\text{Total number of trials}}$

Here, a trial is any action that results in one or more outcomes. An event is a specific collection of these outcomes that we want to track. The value of probability always ranges from 0 to 1, represented mathematically as $0 \le P(E) \le 1$. An event with a probability of 0 is an impossible event, and an event with a probability of 1 is a sure event.

How to Calculate Empirical Probability

1. Identify Total Trials — Examine the problem to find the total number of attempts, trials, tosses, or people surveyed. This total represents your denominator.
2. Identify Favorable Outcomes — Determine the exact frequency of the specific outcome you want to measure. This represents your numerator.
3. Apply the Formula — Write down the probability formula: P(Event) = (Favorable Trials) / (Total Trials). Writing the formula secures step marks in CBSE board exams.
4. Simplify and Calculate — Substitute the values, reduce the fraction to its lowest terms, or convert it to a decimal.

NCERT Style Worked Examples

• Example 1: The Cricket Match Problem In a cricket match, a batswoman hits a boundary 6 times out of 30 balls she plays. Find the probability that she did not hit a boundary. Step-by-step Solution: 1. Total number of balls played (trials) = $30$. 2. Number of times she hits a boundary = $6$. 3. Number of times she does not hit a boundary = Total balls - Boundary balls = $30 - 6 = 24$. 4. Let $E$ be the event of not hitting a boundary. Then, $P(E) = \frac{\text{Number of times she did not hit a boundary}}{\text{Total number of balls played}} = \frac{24}{30} = \frac{4}{5} = 0.8$. Final Answer: The probability that she did not hit a boundary is $0.8$ (or $4/5$).
• Example 2: Two Coin Tosses Two coins are tossed simultaneously 500 times with the following frequencies of outcomes: - Two heads: 105 times - One head: 275 times - No head: 120 times Find the probability of occurrence of each of these events. Step-by-step Solution: 1. Total number of coin tosses (trials) = $500$. 2. Let $E_1$, $E_2$, and $E_3$ be the events of getting two heads, one head, and no head respectively. 3. $P(E_1) = \frac{105}{500} = 0.21$. 4. $P(E_2) = \frac{275}{500} = 0.55$. 5. $P(E_3) = \frac{120}{500} = 0.24$. Verification: Let's add them up: $P(E_1) + P(E_2) + P(E_3) = 0.21 + 0.55 + 0.24 = 1.00$. Since the sum is 1, our calculations are correct.

Pro-Tips & Avoid Common Mistakes

• Always write the formula: CBSE examiners allocate half a mark just for stating the probability formula clearly. Do not skip this!
• Simplify to lowest terms: Never leave your answers as unsimplified fractions like $\frac{142}{200}$. Reduce it to $\frac{71}{100}$ or calculate the decimal equivalent ($0.71$).
• Check your range: Probability can never be negative (e.g., $-0.2$) or greater than 1 (e.g., $1.5$). If you calculate a value outside $[0, 1]$, re-check your numerator and denominator!
• Read carefully: Watch out for negative phrases like 'did not hit', 'dislike', or 'at least'. These change your favorable trial calculation dramatically.

Practice Questions with Solutions

• Q: A coin is tossed 1000 times with the following frequencies: Head: 455, Tail: 545. Compute the probability for each event. A: Step 1: Write down total number of trials. Total trials = 1000. Step 2: Probability of getting a Head, P(H) = Frequency of Head / Total trials = 455 / 1000 = 0.455. Step 3: Probability of getting a Tail, P(T) = Frequency of Tail / Total trials = 545 / 1000 = 0.545. Step 4: Check if P(H) + P(T) = 0.455 + 0.545 = 1.00. Final answer: P(Head) = 0.455, P(Tail) = 0.545.
• Q: A die is thrown 1000 times with the frequencies for the outcomes 1, 2, 3, 4, 5 and 6 given in the following table: Outcome 1: 179, Outcome 2: 150, Outcome 3: 157, Outcome 4: 149, Outcome 5: 175, Outcome 6: 190. Find the probability of getting a number greater than 4. A: Step 1: Identify total number of trials = 1000. Step 2: Identify outcomes greater than 4. These outcomes are 5 and 6. Step 3: Sum the frequencies for outcomes 5 and 6: 175 + 190 = 365. Step 4: Use the formula: P(Number > 4) = 365 / 1000 = 0.365. Final answer: The probability of getting a number greater than 4 is 0.365.
• Q: Out of 50 students in a class, 30 like mathematics while 20 dislike it. Find the probability that a student chosen at random dislikes mathematics. A: Step 1: Identify total trials (total students) = 50. Step 2: Find favorable trials (students who dislike math) = 20. Step 3: Calculate the probability. P(Dislikes) = 20 / 50. Step 4: Reduce to lowest terms: 2 / 5 = 0.4. Final answer: The probability that a student chosen at random dislikes mathematics is 0.4 (or 2/5).
• Q: In a survey of 200 ladies, it was found that 142 like coffee while 58 dislike it. Find the probability that a lady chosen at random likes coffee. A: Step 1: Identify total ladies surveyed = 200. Step 2: Identify favorable outcomes (ladies who like coffee) = 142. Step 3: Apply probability formula. P(Likes coffee) = 142 / 200. Step 4: Simplify to lowest terms: 142 / 200 = 71 / 100 = 0.71. Final answer: The probability that a lady chosen at random likes coffee is 0.71.

Frequently Asked Questions