Welcome to our introduction to probability! Probability measures how likely an event is to happen. It's expressed as a number between 0 and 1. A probability of 0 means an event is impossible, 0.5 means it's equally likely to happen or not happen, and 1 means it's certain to occur. For example, the sun rising tomorrow has a probability close to 1, getting heads on a coin flip is 0.5, and rolling a 6 on a die is about one-sixth or 0.16.
Now let's learn how to calculate probability. The basic formula is: Probability equals the number of favorable outcomes divided by the total number of possible outcomes. Let's use the example of rolling a die. The sample space, which is all possible outcomes, consists of the numbers 1 through 6. So the total number of outcomes is 6. If we want to find the probability of rolling an even number, the favorable outcomes are 2, 4, and 6. That's 3 favorable outcomes. So the probability is 3 divided by 6, which equals one-half or 0.5 or 50%. This means rolling an even number is equally likely as not rolling an even number.
Let's explore probability using a standard deck of playing cards. A deck has 52 cards total, divided into 4 suits: hearts, diamonds, clubs, and spades. Each suit has 13 cards: Ace, numbers 2 through 10, and three face cards - Jack, Queen, and King. Hearts and diamonds are red, while clubs and spades are black. Now let's solve some probability problems. What's the probability of drawing a heart? There are 13 hearts in a deck of 52 cards, so the probability is 13 divided by 52, which equals one-fourth or 25%. What about drawing a face card? There are 12 face cards total (Jack, Queen, and King in each of the four suits), so the probability is 12 divided by 52, which equals three-thirteenths or about 23%.
Now let's learn about complementary events. Complementary events are events that cannot happen at the same time. If one occurs, the other cannot. The sum of their probabilities always equals 1. The formula is: the probability of 'not A' equals 1 minus the probability of A. For example, if the probability of rain tomorrow is 30%, then the probability it won't rain is 1 minus 0.3, which equals 0.7 or 70%. This concept is useful in many applications, such as finding the probability of 'at least one' success, calculating the probability of failure, and solving more complex probability problems. As you can see in our diagram, as the probability of event A increases, the probability of 'not A' decreases, and their sum always equals 1.
Let's summarize what we've learned about probability. Probability measures the likelihood of an event occurring, with values ranging from 0 for impossible events to 1 for certain events. The basic formula for calculating probability is the number of favorable outcomes divided by the total number of possible outcomes. The sample space represents all possible outcomes of an experiment. For complementary events, the probability of 'not A' equals 1 minus the probability of A. Understanding probability helps us make predictions and informed decisions based on data. These concepts form the foundation of probability theory and are essential for further study in statistics and data analysis.