Question

Write and prove more application of basic probability theorem 0

Answer 1

Probability Basics

Probability is defined as a number between 0 and 1 representing the likelihood of an event happening. A probability of 0 indicates no chance of that event occurring, while a probability of 1 means the event will occur. If you're working on a probability problem and come up with a negative answer, or an answer greater than 1, you've made a mistake! Go back and check your work.

Visualizing Probability

There are a number of ways to visualize probabilities, but the easiest way to think about them is to use the fraction method: turn the terms into a fraction by dividing the number of desirable outcomes by the total number of possible outcomes. This will always give you a number between 0 and 1. For example, what are the chances of rolling an odd number on a 6-sided die? There are a total of six numbers and three odd numbers: 1, 3 and 5. So the probability of rolling an odd number is 3/6 or 0.5. You can use this formula when performing more difficult calculations, as we'll see later in the lesson.

In this formula:

P(A) is read as 'the probability of A', where Ais an event we are interested in.

P(A|B) is read as 'the probability of A given Boccurs'.

P(not A) is read as 'the probability of not A ', or 'the probability that A does not occur'.

Probability Rules

There are three main rules associated with basic probability: the addition rule, the multiplication rule, and the complement rule. You can think of the complement rule as the 'subtraction rule' if it helps you to remember it.

1.) The Addition Rule: P(A or B) = P(A) + P(B) - P(A and B)

If A and B are mutually exclusive events, or those that cannot occur together, then the third term is 0, and the rule reduces to P(A or B) = P(A) + P(B). For example, you can't flip a coin and have it come up both heads and tails on one toss.

2.) The Multiplication Rule: P(A and B) = P(A) * P(B|A) or P(B) * P(A|B)

If A and B are independent events, we can reduce the formula to P(A and B) = P(A) * P(B). The term independent refers to any event whose outcome is not affected by the outcome of another event. For instance, consider the second of two coin flips, which still has a .50 (50%) probability of landing heads, regardless of what came up on the first flip. What is the probability that, during the two coin flips, you come up with tails on the first flip and heads on the second flip?

Let's perform the calculations: P = P(tails) * P(heads) = (0.5) * (0.5) = 0.25