Probability Theory: Addition and Multiplication Laws Probability theory provides a framework for analyzing the long-term frequency of certain events and outc...
Probability theory provides a framework for analyzing the long-term frequency of certain events and outcomes in a random experiment. This allows us to make predictions and draw conclusions about the outcomes of repeated trials.
Addition Law:
When an event can occur in multiple ways, the probability of it occurring is the sum of the probabilities of each individual way it can happen.
For example, consider rolling a 6 on a standard six-sided die. The probability of rolling a 6 is 1/6, since there is only one way for a 6 to appear on the die.
Similarly, the probability of rolling a 2 and a 4 on different rolls would be 1/36 + 1/36 = 1/18, as there are two ways to achieve this outcome.
Multiplication Law:
The probability of an event occurring is the product of the probabilities of each individual way it can happen.
For instance, consider rolling a 6 and then rolling a 2 on different rolls. The probability of rolling a 6 and a 2 is equal to the probability of rolling a 6 first, followed by a 2, which is 1/6 x 1/2 = 1/12.
This law applies to situations where there are multiple possible events that can occur independently, with each event having its own probability.
Examples:
Rolling a 5 or 6 on a standard six-sided die.
Getting a head or tail in a single coin toss.
Rolling the same number on two dice.
Key Points:
Probability is a measure of the likelihood of an event occurring.
The addition law tells us that the probability of an event occurring is the sum of the probabilities of all the ways it can happen.
The multiplication law tells us that the probability of an event occurring is the product of the probabilities of all the individual ways it can happen