The document discusses empirical probability, which is determined using the relative frequency of events occurring in trials, emphasizing the law of large numbers. It explains how increasing the number of trials leads to a more accurate representation of probability, illustrated through examples such as coin tosses and student grades. Additionally, it applies this concept to real-world scenarios like life insurance probability assessments and the success rates of space shuttle missions.

1. PRESENTATION TOPIC : EMPIRICAL PROBABILITY

2. EMPIRICAL PROBABILITY.
 Empirical or relative frequency is the second type of objective probability. It is
based on the number of times an event occurs as a proportion of a known
number of trials.
 Definition:
 In terms of a formula:
 Empirical probability =
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑒𝑣𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑠
𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
.
EMPIRICAL PROBABILITY: The probability of an event happening is the fraction
of the time similar events happened in the past.

3. THE EMPIRICAL APPROACH TO PROBABILITY IS BASED ON WHAT IS CALLED THE LAW OF
LARGE NUMBERS. THE KEY TO ESTABLISHING PROBABILITIES EMPIRICALLY IS THAT MORE
OBSERVATIONS WILL PROVIDE A MORE ACCURATE ESTIMATE OF THE PROBABILITY.
 To explain the law of large numbers, suppose we toss a fair coin. The result of each
toss is either a head or a tail. With just one toss of the coin the empirical probability
for heads is either zero or one. If we toss the coin a great number of times, the
probability of the outcome of heads will approach .5. The following table reports the
results of an experiment of flipping a fair coin 1, 10, 500, 1,000, and 10,000 times and
then computing the relative frequency of heads.
 Note: as we increase the number of trials the empirical probability of a head
appearing approaches 0.5, which is its value based on the classical approach to
probability.
LAW OF LARGE NUMBERS Over a large number of trials, the empirical probability
of an event will approach its true probability.

4. TABLE:
Number of Trials
Relative
Number of Heads Frequency of Heads
1 0 0.00
10 3 0.30
500 236 0.472
1,000 494 0.494
10,000 5,027 0.5027

5. EXAMPLES:
 Last semester, 80 students registered for Business Statistics at University. Twelve
students earned an A. Based on this information and the empirical approach to
assigning a probability, we estimate the likelihood a student will earn an A is
0.15.
 Life insurance companies rely on past data to determine the acceptability of an
applicant as well as the premium to be charged. Mortality tables list the likelihood a
person of a particular age will die within the upcoming year. For example, the
likelihood a 20-year-old female will die within the next year is 0.00105.

6. EXAMPLE
On February 1, 2003, the Space Shuttle Columbia exploded. This was the second disaster in
113 space missions for NASA. On the basis of this information, what is the probability that a
future mission is successfully completed?

 Solution:
 To simplify, letters or numbers may be used. P stands for probability and in this Case P(A) stands for the probability a
future mission is successfully completed.
 Probability of a successful flight =
Number of successful flights
Total number of flights
= P(A) =
111
113
= 0.98
 We can use this as an estimate of probability. In other words, based on past experience, the
probability is 0.98 that a future space shuttle mission will be safely completed.

7. Thank you….