Empirical Probability

Empirical probability, also known as experimental probability, is the probability of an event happening based on actual trials or experiments rather than theoretical calculations. It’s determined by observing the outcomes of real-life experiments.

In simple terms, empirical probability is the likelihood of an event based on what has actually happened during multiple trials.

1.0What is Probability?

Probability is a method of measuring how likely an event is to occur. In simple terms, it's the chance of something happening. For example, when you toss a fair coin, there are 2 possible outcomes: heads or tails. The probability\chances of getting heads is 1/2 or 50%, meaning it has an equal chance of happening as getting tails. 

2.0What is Empirical (Experimental) Probability?

Empirical probability, also known as experimental probability, is the probability\likelihood of an event based on actual trials or experiments. Unlike theoretical probability, which is based on assumptions or predictions, empirical probability uses real data from experiments to determine how likely something is to occur.

For example, imagine you want to know the probability\likelihood of getting a 5 when rolling a die. Instead of just predicting the probability based on the fact that a fair die has six faces (which would give a theoretical probability of 1/6), you conduct an experiment by rolling the die several times and recording how many times a 5 appears.

3.0Formula for Empirical Probability

Mathematically, the formula for empirical probability is:

$P(Event)=\frac{Numberoftimestheeventoccurs}{Totalnumberoftrials}$

Where:

• P(Event) is the probability\likelihood of the event happening.
• The numerator is the number of times the event actually occurs in the experiment.
• The denominator is the total number of trials or experiments.

4.0Example of Empirical Probability

Example 1:

A fair die is rolled 120 times. We want to find the number of times the number 5 turns up.

Solution:

The theoretical probability of getting a 5 on a fair die is $\frac{1}{6}$ . However, in an actual experiment, we roll the die 120 times and count how many times we get a 5.

If we find that 5 shows up 20 times out of the 120 rolls, the empirical probability of getting a 5 is: $P(5)=\frac{20}{120}=\frac{1}{6}$

This matches the theoretical probability, which shows that as the number of trials increases, the experimental and theoretical probabilities tend to be closer to each other.

Example 2:

A coin is tossed successively 3 times. Find the probability of getting exactly one head or two heads.

Solution: 

Let S be the sample space and E be the event of getting exactly one head or exactly two heads, then

S = {HHH, HHT, HTH, THH, TTH, THT, HTT, TTT} and E = {HHT, HTH, THH, HTT, THT, TTH}

$\therefore$ n(E) = 6 and n(S) = 8.

Now required probability,

$P(E)=\frac{n(E)}{n(S)}=\frac{6}{8}=\frac{3}{4}$

Example 3:

Words are formed with the letters of the word PEACE. Find the probability that 2 E's come together.

Solution: 

Total number of words which can be formed with the letters P, E, A, C, E $=\frac{5!}{4!}=60$

Number of words in which 2 E’ s come together $=4!=24$

$\therefore required.probability==\frac{24}{60}=\frac{2}{5}$

Example 4:

A bag contains 5 red and 4 green balls. Four balls are drawn at random, then find the probability that two balls are of red and two balls are of green colour.

Solution: 

n(s) = the total number of ways of drawing 4 balls out of total 9 balls: ${}^9{C}_4$

A: Drawing 2 red and 2 green balls;

$n(A)={}^5{C}_2\times {}^4{C}_2$

$\therefore P(A)=\frac{n(A)}{n(s)}={\frac{{}^5}{C}}_2\times {}^4{C}_2{}^9{C}_4=\frac{\frac{5\times 4}{2\times 1}\times \frac{4\times 3}{2\times 1}}{\frac{9\times 8\times 7\times 6}{4\times 3\times 2\times 1}}=\frac{10}{21}$