Statistics and probability FAQ

Basic probability is the chance of something happening, based on what we know or assume. We can find basic probability by using fractions, decimals, or percents. For example, if we toss a fair coin, we know that there are two possible outcomes: heads or tails. Each outcome has the same chance of happening, so we can say that the probability of getting heads is $\dfrac{1}{2}$start fraction, 1, divided by, 2, end fraction, $0.5$0, point, 5, or $50\%$50, percent. We can write this as $P(\text{heads})=\dfrac{1}{2}$P, left parenthesis, start text, h, e, a, d, s, end text, right parenthesis, equals, start fraction, 1, divided by, 2, end fraction, $P(\text{heads})=0.5$P, left parenthesis, start text, h, e, a, d, s, end text, right parenthesis, equals, 0, point, 5, or $P(\text{heads})=50\%$P, left parenthesis, start text, h, e, a, d, s, end text, right parenthesis, equals, 50, percent.

To find the probability of any event, we need to know how many possible outcomes there are, and how many of those outcomes are favorable to the event. When all of the outcomes are equally likely, then the probability of an event is the ratio of the number of favorable outcomes to the number of possible outcomes. For example, if we roll a fair six-sided die, there are six possible outcomes: $1$1, $2$2, $3$3, $4$4, $5$5, $6$6. If we want to find the probability of rolling an even number, we need to count how many of the outcomes are even: $2$2, $4$4, $6$6. There are three favorable outcomes, so the probability of rolling an even number is $\dfrac{3}{6}$start fraction, 3, divided by, 6, end fraction, $0.5$0, point, 5, or $50\%$50, percent. We can write this as $P(\text{even})=\dfrac{3}{6}$P, left parenthesis, start text, e, v, e, n, end text, right parenthesis, equals, start fraction, 3, divided by, 6, end fraction, $P(\text{even})=0.5$P, left parenthesis, start text, e, v, e, n, end text, right parenthesis, equals, 0, point, 5, or $P(\text{even})=50\%$P, left parenthesis, start text, e, v, e, n, end text, right parenthesis, equals, 50, percent.

Try it yourself with our Simple probability exercise.

A probability model is a way of showing all the possible outcomes of a situation and their probabilities. We can make a probability model by using a table, a list, or a tree diagram.

Since the probability model shows all of the possible outcomes, their probabilities will add up to exactly $1$1 (which is the same as $100\%$100, percent).

For example, if we toss two coins, we can make a table like this:

This table shows that there are four possible outcomes: HH, HT, TH, TT. Each outcome has the same probability of $\dfrac{1}{4}$start fraction, 1, divided by, 4, end fraction, $0.25$0, point, 25, or $25\%$25, percent. We can use this table to answer questions like: What is the probability of getting two heads? What is the probability of getting at least one tail? What is the probability of getting the same face on both coins?

Try it yourself with our Probability models exercise.

A compound event is an event that involves two or more simple events. For example, rolling a die and tossing a coin is a compound event. A sample space is the set of all possible outcomes of a compound event. For example, the sample space for rolling a $6$6-sided die and tossing a coin is:

To build the sample space, we paired every outcome from the first event with every outcome from the second event. In the table, each row represents a different overall outcome.

This sample space shows that there are $12$12 possible outcomes for this compound event. We can use the sample space to find the probability of any event that involves both the die and the coin.

For example, what is the probability of rolling a $3$3 and getting a head? We can see that there is only $1$1 outcome that matches this event: $3$3, Heads. So the probability of this event is $\dfrac{1}{12}$start fraction, 1, divided by, 12, end fraction, about $0.083$0, point, 083, or about $8.3\%$8, point, 3, percent.

On the other hand, there are $2$2 outcomes for rolling a number greater than $4$4 and getting tails: $5$5, tails; and $6$6, tails. So the probability of that event is $\dfrac{2}{12}$start fraction, 2, divided by, 12, end fraction, which is equal to $\dfrac{1}{6}$start fraction, 1, divided by, 6, end fraction.

Try it yourself with our Probability models exercise.

Try it yourself with our Sample spaces for compound events exercise and Probabilities of compound events exercise.

Sampling populations is the process of selecting a small group of people or things from a larger group, called a population, and using the sample to learn something about the population.

For example, if we want to know how many students in our school like pizza, we often cannot ask every student in the school. That would take too long and be impractical. Instead, we can select a sample of students, such as a random class from each grade level, and ask them if they like pizza. Then we can use the results from the sample to estimate the results for the whole population.

Taking random samples of populations is important because it helps us make inferences or generalizations about a large group based on a small group. However, sampling populations also has some challenges and limitations. We need to make sure that our sample is representative, meaning that it reflects the diversity and characteristics of the population. We also need to be aware of bias, meaning that something influences or skews the results of the sample. For example, if we only sample students who are in the cafeteria during lunch, we might get a biased result, because they are more likely to like pizza than students who are not in the cafeteria.

Try it yourself with our Valid claims exercise.