What is Probability?
Probability measures how likely it is for an event to occur. For example, determining the chance of a coin landing on heads or tails when you toss it.
Key Terms in Probability
- Sample Space: The set of all possible outcomes from an experiment. For a coin toss, the sample space is {Heads, Tails}.
- Event: A specific outcome that we’re interested in. For instance, getting “Heads” when tossing a coin.
How to Calculate Probability?
The probability of an event is calculated by the formula:
Probability (P) = Number of favorable outcomes / Total number of outcomes
This calculates how likely it is for an event to happen based on the total number of possible outcomes.
Fundamental Probability Concepts
- Mutually Exclusive Events – Two events are mutually exclusive if they cannot happen at the same time. For instance, when rolling a single die, getting a 2 and a 5 simultaneously is impossible.
Example: If you roll a six-sided die, the probability of rolling either a 2 or a 5 is:
P(2 or 5) = P(2) + P(5) = 1/6 + 1/6 = 1/3
- Independent Events are independent if the occurrence of one does not affect the occurrence of the other. For example, flipping a coin and rolling a die simultaneously are independent events.
Example: The probability of flipping a coin and getting heads, then rolling a die and getting a 6, is:
P(Heads and 6) = P(Heads) x P(6) = 1/2 x 1/6 = 1/12
- Conditional Probability measures the probability of an event occurring, given that another event has already occurred.
Example: If a card drawn is a king, what is the probability it is red?
Given there are 2 red kings in a standard deck of 52 cards:
P(Red given King) = Number of Red Kings / Number of Kings = 2/4 = 1/2
- Bayes’ Theorem is used to update probabilities with new evidence. It is particularly useful in calculating conditional probabilities.
Example: Assuming a disease affects 1 in 1,000 people, and a test for the disease is 99% accurate, calculate the probability that a person has the disease if they test positive.
Let:
- P(Disease) = 0.001 (prevalence of the disease)
- P(No Disease) = 0.999 (likelihood of not having the disease)
- P(Positive | Disease) = 0.99 (probability of testing positive if having the disease)
- P(Positive | No Disease) = 0.01 (probability of testing positive without having the disease)
Using Bayes’ Theorem:
P(Disease | Positive) = (P(Positive | Disease) x P(Disease)) / P(Positive)
First, calculate P(Positive):
P(Positive) = P(Positive | Disease) x P(Disease) + P(Positive | No Disease) x P(No Disease) = 0.99 x 0.001 + 0.01 x 0.999 = 0.01098
Then, substitute back to find P(Disease | Positive):
P(Disease | Positive) = (0.99 x 0.001) / 0.01098 ≈ 0.09016
This guide provides a basic understanding of probability through examples, helping you see how probability is applied in different scenarios. These concepts are essential in fields like business, science, and healthcare, enabling better decision-making based on likely outcomes.