What is a Probability Function?It is a function that associates each possible outcome of a discrete random variable with its probability of occurrence.Simple example: Rolling a fair die.- Random variable (X): Number on the face (1, 2, 3, 4, 5, 6).- Probability of each face: \( P(X = x) = \frac{1}{6} \).The probability function is the table or formula that shows this association.To be valid, the function must satisfy:1. Non-negativity:\( P(X = x) \geq 0 \) for every \( x \).(Example: There is no negative probability, such as -0.2 for rolling the number 5 on the die.)2. Sum equal to 1:\( \sum_{\text{all } x} P(X = x) = 1 \).(Example: \( \frac{1}{6} + \frac{1}{6} + \dots + \frac{1}{6} = 1 \)).3. Difference for Continuous Variables- In discrete variables (such as dice, coins), we use a probability function (\( P(X = x) \)).- In continuous variables (such as height, weight), we use a probability density function (p.d.f.), since \( P(X = x) = 0 \) for any specific \( x \).Bernoulli DistributionThe Bernoulli Distribution is used to model situations in which there are only two possible outcomes: success (1) or failure (0). Each trial is independent, and the probabilities of success (\( p \)) and failure (\(q\)) are constant.Formula of the Bernoulli DistributionThe probability of a Bernoulli event is given by:\[ P(X = 1) = 1 - P(X = 0) = 1 - q = p \]Where:- \( k = 1 \) (success) or \( k = 0 \) (failure).- \( p \): Probability of success.- \( q \): Probability of failure (also written as \( 1 - p \)).Since in every Bernoulli event we are looking for only 1 or 0 successes (\( n \) being the total number of trials, 1 in Bernoulli, and \( k \) being the total number of successes). Thus, we can also write the probability as:\[ P(X = k) = p^k \cdot (1 - p)^{n - k} \]Real-Life ExamplesExample 1: Tossing a coin- Success: Heads (\( k = 1 \)).- Failure: Tails (\( k = 0 \)).- Probability of success (\( p \)): 0.5.Question:What is the probability of getting heads in a single toss?Solution:\[ P(X = 1) = 0.5^1 \cdot (1 - 0.5)^{1 - 1} = 0.5 \quad (50\%) \]Example 2: Guessing a multiple-choice question- Success: Getting it right (\( k = 1 \)).- Failure: Getting it wrong (\( k = 0 \)).- Probability of success (\( p \)): 0.25 (4 options).Question:What is the probability of getting the question right?Solution:\[ P(X = 1) = 0.25^1 \cdot (1 - 0.25)^{1 - 1} = 0.25 \quad (25\%) \]Binomial DistributionThe Binomial Distribution is used to model the number of successes in \( n \) independent trials of a Bernoulli experiment, where each trial has the same probability of success \( p \).Formula of the Binomial DistributionThe probability of getting exactly \( k \) successes in \( n \) trials is given by:\[ P(X = k) = C_{n,k} \cdot p^k \cdot (1 - p)^{n - k} \]Where:- \( C_{n,k} \): Combination of \( n \) elements taken \( k \) at a time.- \( p \): Probability of success in a single trial.- \( 1 - p \): Probability of failure.- \( n \): Total number of trials.- \( k \): Number of desired successes.Graph representing the probabilities in the Binomial distributionReal-Life ExamplesExample 1: Tossing a coin 5 times- Success: Heads (\( k = 1 \)).- Failure: Tails (\( k = 0 \)).- Probability of success (\( p \)): 0.5.- Number of trials (\( n \)): 5.Question:What is the probability of getting exactly 3 heads?Solution:\[ C_{5,3} = \frac{5!}{3! \cdot (5 - 3)!} = 10 \]\[ P(X = 3) = 10 \cdot 0.5^3 \cdot (1 - 0.5)^{5 - 3} = 10 \cdot 0.125 \cdot 0.25 = 0.3125 \quad (31.25\%) \]Example 2: Getting 2 questions right in 5 guesses- Success: Getting it right (\( k = 1 \)).- Failure: Getting it wrong (\( k = 0 \)).- Probability of success (\( p \)): 0.25 (4 options).- Number of trials (\( n \)): 5.Question:What is the probability of getting exactly 2 questions right?Solution:\[ C_{5,2} = \frac{5!}{2! \cdot (5 - 2)!} = 10 \]\[ P(X = 2) = 10 \cdot 0.25^2 \cdot (1 - 0.25)^{5 - 2} = 10 \cdot 0.0625 \cdot 0.421875 = 0.2637 \quad (26.37\%) \]Example 3: Rolling a die 4 times- Success: Getting the number 6 (\( k = 1 \)).- Failure: Not getting the number 6 (\( k = 0 \)).- Probability of success (\( p \)): \( \frac{1}{6} \).- Number of trials (\( n \)): 4.Question:What is the probability of getting the number 6 exactly 2 times?Solution:\[ C_{4,2} = \frac{4!}{2! \cdot (4 - 2)!} = 6 \]\[ P(X = 2) = 6 \cdot \left(\frac{1}{6}\right)^2 \cdot \left(\frac{5}{6}\right)^{4 - 2} = 6 \cdot \frac{1}{36} \cdot \frac{25}{36} \approx 0.1157 \quad (11.57\%) \]Exercises1. Tossing a coin 8 times:- What is the probability of getting exactly 5 heads?- What is the probability of getting fewer than 3 heads?2. Guessing 10 multiple-choice questions (4 options):- What is the probability of getting exactly 4 questions right?- What is the probability of getting at least 5 questions right?3. Rolling a die 6 times:- What is the probability of getting the number 5 exactly 2 times?- What is the probability of getting the number 5 at most 3 times?4. A soccer team has a 70% chance of winning each match. In 5 games:- What is the probability of winning exactly 3 matches?- What is the probability of winning at least 4 matches?Answer Key1. Tossing a coin 8 times:- 21.88%- 14.45%2. Guess 10 questions:- 14.60%- 7.81%3. Roll a die 6 times:- 20.09%- 99.13%4. Soccer team:- 30.87%- 52.82%
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