The ratio of the number of ways an event can occur to the number of possible outcomes

Probability refers to the chance that an event will happen.

Theoretical Probability of Event A

Inhaltsverzeichnis Show

• What is the ratio of the number of ways a specific event can occur to the total number of possible outcomes?
• What probability is the ratio of the number of outcomes in an event to the number of members in the sample space?
• Is the ratio of the number of times an event occurs to the total number of trials or times the activity is performed?
• What terms refer to the ratio of the number of times an event occurs to the number of trials?

Probability can be presented as a ratio of the number of ways an event can occur relative to the number of possible outcomes.

Example 1:

If a fair-sided die is rolled, what is the probability of rolling a four?

Solution:

The possible outcomes are rolling a one, two, three, four, five, or six.

So, there are six possible outcomes.

There is one way the event of rolling a four can occur.

Example 2:

If a fair-sided die is rolled, what is the probability of rolling a number less than four?

Solution:

The possible outcomes are rolling a one, two, three, four, five, or six.

There are three rolls which could produce the event of rolling a number less than four.

Complement refers to the chance that an event will not happen.

Example 3:

If a fair-sided die is rolled, what is the probability that the result is NOT less than three?

Solution:

The possible outcomes are rolling a one, two, three, four, five, or six.

First, find the probability of rolling a number less than three.

There are two rolls out of six which could produce the event of rolling a number less than three.

Now, find the probability of rolling a number NOT less than three.

In general, events A and B are dependent if the occurrence of A affects the probability of B occurring, and the occurrence of B affects the probability of A occurring.

For any two dependent events, A and B, the following statements are true.

Example 4:

A bag of chips contains two blue chips and four red chips. Thom draws two chips from the bag without replacement. What is the probability of drawing a blue chip on the first draw and a blue chip on the second draw?

Solution:

Since the first chip is not replaced, there will be one less chip overall in the bag for the second draw. There will also be one less blue chip since the first chip that was drawn was blue. In this case, the first draw affected the probability of the second draw, so the two events are dependent.

In general, events A and B are independent if the occurrence of A does not affect the probability of B occurring, and the occurrence of B does not affect the probability of A occurring.

If event A and event B are independent, the probability of both event A and event B occurring is as follows.

Example 5:

If a coin is flipped twice, what is the probability that the coin lands on heads both times?

Solution:

First, find the probability of each event.

Since P(heads on 1st flip) and P(heads on 2nd flip) are independent events, the probability that the coin lands on heads both times is shown below.

If event A and event B are independent, the probability of event A or event B occurring is as follows. 

Example 6:

If a fair-sided die is rolled, what is the probability that a two or a five is rolled?

Solution:

First, find the probability of each event.

Since P(2) and P(5) are independent events, the probability that a two or a five is rolled is shown below.

Experimental Probability of Event A

An experimental probability is based on data collected from an experiment.

Example 7:

Joey tossed a coin ten times and recorded his results in the table below.

What is the experimental probability of the coin landing on heads?

Solution:

There were a total of ten outcomes and six of them were heads.

Therefore, the experimental probability of the coin landing on heads is , which is equal to .

Example 8:

A spinner has eight equal-sized sections. Two sections are yellow, two are green, two are blue, one is red, and one is purple. J.D. spun the spinner twelve times and recorded the results in the table below.

Which color's experimental probability matches its theoretical probability?

Solution:

First, calculate the theoretical probability of spinning yellow.

Now, calculate the experimental probability of spinning yellow.

Therefore, the experimental probability of spinning yellow is equal to the theoretical probability of spinning yellow.

Given events A and B, the conditional probability that event B will occur, given that event A has already occurred, can be found using the following formula.P(A) is the probability that event A will occur.

P(A  B) is the probability that event A and event B will occur.

Example 9:

David ordered 18 pizzas for a party. There are six pizzas that have pepperoni and sausage, and there are nine pizzas that have pepperoni only. What is the probability that a pizza has sausage, given that it also has pepperoni?

Solution:

Define the events.

The question asks for the probability that a pizza has sausage given that the pizza has pepperoni; therefore, the formula for conditional probability must be applied.

Rewrite the formula for conditional probability for the given events, P and S.

Find P(P). There are 18 pizzas, and 15 of the pizzas have pepperoni.

Find P(P  S). There are 18 pizzas, and 6 of the pizzas have pepperoni and sausage.

Substitute P(P) and P(P 

A permutation of a set of n distinct objects taken r at a time is an arrangement of the r objects in a specific order without repetition.In a permutation, AB does not equal BA. The formula used with permutations, where order does matter, is shown below.A combination of a set of n distinct objects taken r at a time without repetition is an r element subset of the set of n objects. In a combination, AB is equal to BA. The formula used with combinations, where order does not matter, is shown below.

Example 10:

Sarah is attending a photography contest at an art gallery. The gallery will award five different prizes to the top five winners of the contest. Sarah decides to predict the top five winners and the prizes that they will receive. If there are 20 entries in the contest, and each entry is equally like to win, what is the probability that Sarah will guess correctly?

Solution:

Since the prizes are different, the order does matter. Use the formula to calculate the permutations of 20 items taken 5 at a time.

Only one of these possibilities is what Sarah predicted.

Example 11:

Mr. William teaches seventh grade and is the sponsor of a school club, which consists of 12 students. Mr. William is forming a team of four club members to organize a special club project. A club member, Aaron, decides to predict the club members that Mr. William will select for the team. If each club member is equally likely to be selected, then what is the probability that Aaron guessed the team members correctly?

Solution:

Since the order of the members does not matter, use the formula to calculate the combinations of 12 items taken 4 at a time.

Only one of these combinations is what Aaron predicted.

What is the ratio of the number of ways a specific event can occur to the total number of possible outcomes?

What probability is the ratio of the number of outcomes in an event to the number of members in the sample space?

Is the ratio of the number of times an event occurs to the total number of trials or times the activity is performed?

What terms refer to the ratio of the number of times an event occurs to the number of trials?