Experimental (Empirical) Probability

Experimental (Empirical) probability is a way of determining the probability of an event occurring by actually performing an experiment or collecting data. Unlike theoretical probability, which is based on logical reasoning and ideal situations, experimental probability relies on the results of real-world trials. It’s a practical approach to estimating the likelihood of an event when you can’t easily calculate it theoretically.

Formulae

The formula for experimental probability is:

The probability of an event E, denoted by $P(E)$, is calculated as:

$P(E) = \frac{\text{Favorable Trials}}{\text{Total Trials}}$

Examples

Example-1: Flipping a Coin

A coin is flipped 50 times. Heads appears 28 times. What is the experimental probability of getting heads?

Solution:

• Favorable Trials (Heads): 28
• Total Trials: 50
• $P(\text{Heads}) = \frac{28}{50} = 0.56$ or 56%

Example-2: Rolling a Die

A six-sided die is rolled 100 times. The number 3 appears 15 times. What is the experimental probability of rolling a 3?

Solution:

• Favorable Trials (Rolling a 3): 15
• Total Trials: 100
• $P(\text{Rolling a 3}) = \frac{15}{100} = 0.15$ or 15%

Common Mistakes by Students

• Confusing with Theoretical Probability: Students sometimes mix up experimental probability (based on experiments) with theoretical probability (based on logic and ideal scenarios). Remember that experimental probability is derived from data collected during actual trials.
• Ignoring Sample Size: A small number of trials can lead to inaccurate estimates. The more trials you conduct, the closer your experimental probability will likely be to the true probability (as the number of trials goes to infinity).
• Miscounting Favorable Outcomes: Carelessly counting the number of favorable outcomes can lead to incorrect results. Always double-check your count.
• Incorrectly Applying the Formula: Students may sometimes reverse the formula or use the wrong numbers in the numerator or denominator. Always remember to divide the number of favorable outcomes by the total number of trials.

Real Life Application

Experimental probability is used in many real-world applications, including:

• Insurance: Insurance companies use experimental probability based on historical data (e.g., accident rates, disease prevalence) to set premiums.
• Market Research: Businesses use surveys and experiments to determine the likelihood of customers purchasing a product or service.
• Quality Control: Manufacturers use experimental probability to assess the rate of defects in their products.
• Sports Analytics: Coaches and analysts use experimental probability based on game statistics to evaluate player performance and predict game outcomes.
• Weather Forecasting: Historical weather data is used to determine the probability of rainfall, sunshine, or other weather events.

Fun Fact

The more trials you perform in an experiment, the closer your experimental probability will get to the theoretical probability, if such a theoretical probability exists. This concept is related to the Law of Large Numbers, a fundamental principle in probability theory.

Recommended YouTube Videos for Deeper Understanding

Q.1 A coin is tossed 100 times, and heads appear 60 times. What is the experimental probability of getting tails?

Check Solution

Ans: A

The number of tails is $100 – 60 = 40$. Therefore, the experimental probability of getting tails is $40/100 = 0.4$.

Q.2 A die is rolled 50 times, and the number 3 appears 8 times. What is the experimental probability of not rolling a 3?

Check Solution

Ans: B

The number of times not rolling a 3 is $50 – 8 = 42$. Therefore, the experimental probability is $42/50 = 0.84$.

Q.3 A bag contains 20 marbles. A student picks a marble, records its color, and replaces it. They do this 30 times and record picking a red marble 12 times. What is the experimental probability of picking a blue marble?

Check Solution

Ans: D

We are given that the red marble was picked 12 times out of 30 trials. We do not have information to determine the frequency of picking a blue marble and hence the probability.

Q.4 A survey of 80 students shows that 32 of them like pizza. Based on this survey, what is the experimental probability that a student likes pizza?

Check Solution

Ans: B

The experimental probability is the number of students who like pizza divided by the total number of students, which is $32/80 = 0.4$.

Q.5 In a game, a player spins a spinner 25 times. The spinner landed on the number 5 a total of 7 times. What is the experimental probability of the spinner *not* landing on the number 5?

Check Solution

Ans: B

The spinner *not* landing on 5 a total of $25-7 = 18$ times. Therefore the experimental probability is $18/25 = 0.72$.

Next Topic: Probability: Basic Terminology