Probability — Telangana (SSC) Class 9 Mathematics Solutions (Free)

Empirical (or experimental) probability is calculated directly from the actual results of a real experiment, rather than from theoretical reasoning about equally likely outcomes: P(event) = (Number of times the event occurred) / (Total number of trials) Given: total trials = 100, heads occurred = 60 times. P(heads) = 60/100 = 3/5 = 0.6 The empirical probability of heads is 0.6, or 60%. Note that this differs from the THEORETICAL probability of 0.5 (50%) expected for a fair coin — a real experiment's result will rarely match the theoretical value exactly, purely due to the randomness of any finite number of trials.

P(5) = (Number of times 5 appeared) / (Total number of rolls) P(5) = 28/120 = 7/30 ≈ 0.233 The empirical probability of rolling a 5 is approximately 0.233, or 23.3%. The theoretical probability for a fair six-sided die would be 1/6 ≈ 0.167 — the experimental result being somewhat higher illustrates that with a limited number of trials (120, not thousands), some deviation from the theoretical value is expected and normal.

P(has a car) = (Number of families with a car) / (Total families surveyed) P(car) = 85/200 = 17/40 = 0.425 The empirical probability is 0.425, or 42.5%. Unlike a coin toss or die roll, there is no separate "theoretical" probability to compare this against — car ownership isn't governed by equally likely physical outcomes, so a real survey like this IS the only way to estimate this probability. This is exactly the kind of situation empirical probability is most essential for: real-world events where a theoretical model doesn't exist or wouldn't apply.

Empirical probability (from the actual trials): P(red)ₑₘₚᵢᵣᵢcₐₗ = 52/100 = 13/25 = 0.52 Theoretical probability (from the known contents of the bag): P(red)ₜₕₑₒᵣₑₜᵢcₐₗ = (red balls) / (total balls) = 5/10 = 1/2 = 0.5 Comparison: the empirical result (0.52) is very close to the theoretical value (0.5), differing by only 0.02. This small difference is expected random variation from a finite number of trials, not a sign that anything is wrong with the bag or the drawing process — with replacement after each draw, drawing "with replacement" 100 times is like tossing a biased coin 100 times, and small deviations from the expected proportion are entirely normal at this sample size.

Empirical probability: P(ace)ₑₘₚᵢᵣᵢcₐₗ = 35/500 = 7/100 = 0.07 Theoretical probability: a standard deck has 52 cards, of which 4 are aces (one per suit). P(ace)ₜₕₑₒᵣₑₜᵢcₐₗ = 4/52 ≈ 0.077 Comparison: the empirical result (0.07) and theoretical value (≈0.077) are approximately equal, differing by less than 0.01. Notice this is a CLOSER match than the previous (bag of balls) example, despite both starting from a random process — this illustrates a general pattern in probability: as the number of trials increases (500 here versus 100 there), the empirical probability tends to converge more closely toward the theoretical probability, a principle known as the Law of Large Numbers.

P(defective) = (Number of defective items) / (Total items in the batch) P(defective) = 8/1000 = 1/125 = 0.008 The empirical probability that an item is defective is 0.008, or 0.8%. This is a genuine, practical use of empirical probability in industry: quality-control teams use exactly this kind of calculation on sample batches to estimate a manufacturing process's overall defect rate, which then informs decisions about acceptable tolerances, inspection frequency, and whether a production line needs adjustment.

Applying P(event) = (occurrences) / (total trials) to each colour: P(Red) = 45/150 = 3/10 = 0.3 P(Blue) = 35/150 = 7/30 ≈ 0.233 P(Green) = 40/150 = 4/15 ≈ 0.267 P(Yellow) = 30/150 = 1/5 = 0.2 Verification: since every single spin must land on exactly one of these four colours, their probabilities must add up to exactly 1 (representing 100% certainty that the spinner lands on SOME colour). Sum = 0.3 + 0.233 + 0.267 + 0.2 = 1.0 ✓ This check — that the probabilities of all possible, non-overlapping outcomes of an experiment always sum to 1 — is a useful way to catch arithmetic errors in any probability problem involving several categories, since a sum that comes out to anything other than 1 signals a mistake somewhere in the calculation.

P(rain) = (Number of rainy days) / (Total days recorded) P(rain) = 120/365 = 24/73 ≈ 0.329 The empirical probability of rain on a given day, based on this year's data, is approximately 0.329, or 32.9%. This is exactly the method meteorologists use to state a "percent chance of rain" — it is fundamentally a statement of EMPIRICAL probability, based on how often rain has historically occurred under similar recorded conditions, not a theoretical calculation from first principles, since weather has no small set of equally likely fixed outcomes to reason about the way a coin or die does.