Empirical and Theoretical Probability

Empirical and theoretical probability are essential concepts in Statistics. Theoretical probability is calculated using known formulas, assuming all outcomes are equally likely—like the chance of rolling a 6 on a fair die. Empirical probability, on the other hand, is based on actual experiments or observed data. It reflects real-world outcomes and may vary with each trial. Understanding both types helps in analyzing patterns, making predictions, and solving problems in fields like science, business, and daily life.

1.0What is Empirical Probability?

Empirical probability, also known as experimental probability, is the probability of an event based on actual data or experiments, rather than theory or assumptions.

It is calculated by observing how often an event occurs in a number of trials or experiments.

2.0Empirical Probability Formula

$\text{Empirical Probability(P)}=\frac{\text{Number of times the event occurs}}{\text{Total number of trials}}$

3.0Empirical Probability Examples with Solutions

Example 1: A coin is tossed 100 times. It lands on heads 56 times. What is the empirical probability of getting a head?

Solution:

$P(Head)=\frac{56}{100}=0.56$

Example 2: A student rolls a die 60 times and observes that the number "4" appears 12 times. Find the empirical probability of rolling a 4.

Solution:

$P(4)=\frac{12}{60}=0.2$

Example 3: A biased coin is tossed 1000 times and it shows heads 430 times. Estimate the empirical probability of getting a tail.

Solution:

Total trials = 1000

Number of heads = 430

Number of tails = 1000 – 430 = 570

$\text{Empirical Probability of tail}=\frac{570}{1000}=0.57$

Example 4: A student rolls a die 600 times and records the following results:

Find the empirical probability of getting a number less than or equal to 3.

Solution:

Favorable outcomes = Frequency of 1 + 2 + 3 = 95 + 100 + 105 = 300

Total trials = 600

$P(\le 3)=\frac{300}{600}=0.5$

Example 5: In a survey conducted among 800 students, 640 preferred Mathematics over Physics. Estimate the empirical probability that a randomly selected student prefers Mathematics.

Solution:

$\text{P(Mathematics)}=\frac{640}{800}=0.8$

4.0What is Theoretical Probability?

Theoretical Probability is the probability that an event will occur based on mathematical reasoning and known outcomes. It assumes that all outcomes are equally likely.

5.0Theoretical Probability Formula

$\text{Theoretical Probability(P)}=\frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$

6.0Theoretical Probability Examples

Example 1: What is the probability of rolling a 3 on a fair 6-sided die?

Solution:

$P(3)=\frac{1}{6}\approx 0.167$

Example 2: A card is drawn from a standard deck of 52 cards. Find the probability of drawing a King.

Solution:

There are 4 Kings in a deck.

$P(King)=\frac{4}{52}=\frac{1}{13}\approx 0.077$

Example 3: A number is selected at random from the first 100 natural numbers. What is the probability that the number is divisible by 6 or 8?

Solution:

• Numbers divisible by 6 = $[\frac{100}{6}]=16$
• Numbers divisible by 8 = $[\frac{100}{8}]=12$
• Numbers divisible by both (LCM = 24) =  $[\frac{100}{24}]=4$

Using inclusion-exclusion:

$$\begin{gathered} n(AUB)=16+12-4=24 \\ \text{Total outcomes}=100 \\ P=\frac{24}{100}=0.24 \end{gathered}$$

Example 4: Two cards are drawn from a well-shuffled deck of 52 cards without replacement. Find the probability that both cards are aces.

Solution:

Favorable outcomes:

• First card: 4 Aces → $\frac{4}{52}$
• Second card: 3 Aces left → $\frac{3}{51}$

$P=\frac{4}{52}\times \frac{3}{51}=\frac{1}{221}\approx 0.00452$

Example 5: A point is selected randomly inside a square of 10 units. What is the probability that the point lies within a circle of radius 5 units inscribed in the square?

Solution:

• Area of square = $10^2=100$
• Area of circle = $\pi r^2=\pi (5{)}^2=25\pi$

$P=\frac{25\pi }{100}=\frac{\pi }{4}\approx 0.7854$

Example 6: A number is selected at random from the set {1, 2, 3, ..., 1000}. What is the probability that it is divisible by both 4 and 6?

Solution:

LCM of 4 and 6 = 12

Count of numbers divisible by 12 = $[\frac{1000}{12}]=83$

$P=\frac{83}{1000}=0.083$ 

Also Check: Probability and Statistics previous year questions with solutions

7.0Difference between Empirical and Theoretical Probability

8.0Practice Questions

1. A spinner was spun 50 times and landed on red 18 times. Find the empirical probability of landing on red.
2. What is the theoretical probability of getting an even number on a fair six-sided die?
3. Out of 200 surveys, 160 people liked a new product. What is the empirical probability that a person likes the product?
4. What is the theoretical probability of picking a vowel from the English alphabet?
5. A coin was tossed 20 times and got tails 13 times. What is the empirical probability of getting tails?

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