Probability of an Event

Understanding Probability

Have you ever wondered about the chances of rain today? Or what's the probability of your favorite team winning a match? In daily life, we often face uncertain situations, and this is where the concept of probability becomes important.

Probability is a measure of the likelihood of an event occurring. Probability gives a numerical value between 0 and 1, where:

• Probability 0 means the event is impossible to occur
• Probability 1 means the event will certainly occur
• Probability 0.5 means the event has an equal chance of occurring or not

The concept of probability helps us make decisions based on data and analysis, not just based on intuition alone.

Sample Space and Events

Before calculating probability, we need to understand two fundamental concepts:

Sample Space (S) is the set of all possible outcomes from an experiment. For example, when rolling two dice, the sample space consists of all possible pairs of numbers that can appear.

Event (A) is a subset of the sample space that represents the outcome we want or the outcome we are examining.

Concrete example: when flipping two coins simultaneously, the sample space is $\{AA, AG, GA, GG\}$, where A = Heads and G = Tails. If we want to know the probability of getting at least one tail, then event A = $\{AG, GA, GG\}$.

Probability Formula

To calculate the probability of an event, we use the classical probability formula:

Where:

• $P(A)$ = probability of event A occurring
• $n(A)$ = number of favorable outcomes (members of event A)
• $n(S)$ = number of all possible outcomes (members of sample space)

Important condition: This formula applies when all outcomes in the sample space have equal probability of occurring (equiprobable).

Fundamental properties of probability:

• $0 \leq P(A) \leq 1$ (probability is always between 0 and 1)
• $P(A) + P(A') = 1$, where $A'$ is the complement of event A

Complement Concept: The complement of event A is all outcomes in the sample space that are not included in event A. For example, if A is "getting an even number", then A' is "getting an odd number".

Applications in Real Situations

Analysis of Rolling Two Dice:

When rolling two dice, the total possible outcomes are $6 \times 6 = 36$ pairs. Let's analyze the probability of getting a sum of 9:

Systematic steps:

1. Identify all ways to get sum 9:

   • First die 3, second die 6: (3,6)
   • First die 4, second die 5: (4,5)
   • First die 5, second die 4: (5,4)
   • First die 6, second die 3: (6,3)

2. Count the number of favorable events: $n(A) = 4$

3. Determine probability:

   $$P(\text{sum} = 9) = \frac{4}{36} = \frac{1}{9} \approx 0.111$$

Probability-Based Marketing Strategy:

A beverage company runs a prize program by inserting coupons in each milk box. Based on historical data, the probability of someone buying a milk box containing a prize is $\frac{3}{32}$.

Practical interpretation: Out of every 32 milk boxes produced, an average of 3 boxes contain prizes. This information helps the company:

• Plan promotional budgets
• Estimate consumer response
• Determine sales targets

Strategic Gaming:

In traditional dice games, players often use probability understanding to make decisions. For example, the probability of getting sum 7 (which occurs most frequently) is:

Ways to get sum 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 ways

This is the highest probability compared to other sums, so it's often used as the basis for game strategy.

Practice Problems

1. In rolling two dice, determine the probability of getting a sum that is an even number.

2. A box contains 5 red balls, 3 blue balls, and 2 green balls. If one ball is drawn randomly, determine the probability of getting a ball that is not red.

3. In flipping three coins simultaneously, determine the probability of getting exactly two tails.

4. A company produces 1000 units of product. Based on experience, 5% of these products are defective. If a consumer buys one product randomly, what is the probability that the product is not defective?

Answer Key

1. Answer: $\frac{1}{2}$

   Systematic solution steps:

   Sample space for rolling two dice: $n(S) = 36$

   Identify all even sums and ways to obtain them:

   • Sum 2: (1,1) → 1 way
   • Sum 4: (1,3), (2,2), (3,1) → 3 ways
   • Sum 6: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 ways
   • Sum 8: (2,6), (3,5), (4,4), (5,3), (6,2) → 5 ways
   • Sum 10: (4,6), (5,5), (6,4) → 3 ways
   • Sum 12: (6,6) → 1 way

   Total favorable events: $n(A) = 1 + 3 + 5 + 5 + 3 + 1 = 18$

   Probability calculation:

   $$P(\text{even}) = \frac{18}{36} = \frac{1}{2} = 0.5$$

2. Answer: $\frac{1}{2}$

   Solution steps:

   Count total balls: $5 + 3 + 2 = 10$ balls

   Identify balls that are not red: $3 \text{ blue} + 2 \text{ green} = 5$ balls

   Probability calculation:

   $$P(\text{not red}) = \frac{5}{10} = \frac{1}{2} = 0.5$$

3. Answer: $\frac{3}{8}$

   Solution steps:

   Sample space for three coin flips: $\{AAA, AAG, AGA, AGG, GAA, GAG, GGA, GGG\}$

   Total possibilities: $n(S) = 2^3 = 8$

   Event exactly two tails: $\{AGG, GAG, GGA\}$

   Number of favorable events: $n(A) = 3$

   Probability calculation:

   $$P(\text{exactly 2 tails}) = \frac{3}{8} = 0.375$$

4. Answer: $\frac{19}{20}$ or 95%

   Solution steps:

   Total products: 1000 units

   Defective products: $5\% \times 1000 = 50$ units

   Non-defective products: $1000 - 50 = 950$ units

   Probability calculation:

   $$P(\text{not defective}) = \frac{950}{1000} = \frac{19}{20} = 0.95 = 95\%$$

   Interpretation: There is a 95% probability that the product purchased by a consumer is not defective.