Counting and Probability

Probability

Probability Definitions

The probability of an event is a number from zero to one (or a percentage from 0% to 100%) that describes how likely the event is to occur.

Probability is the likelihood that an event will happen. The probability of an event is always between zero and one, or between 0% and 100%. An event with a probability of zero is impossible, and an event with a probability of one is certain. Events with a probability closer to zero are unlikely, while those with a probability closer to one are likely. Events with the same probability are equally likely.

The probability of an event can be written as P(event)P(\text{event}), which is read as "the probability of the event." In an experiment that involves flipping a coin, the notation P(heads)P(\text{heads}) or P(H)P(H) may be used to denote the probability that the outcome is heads.

The probabilities of all the outcomes in the sample space of an event have a sum of 1 because it is certain that one of the outcomes will happen. The complement of an event is all outcomes in the sample space that are not the event. For example, if an event (E)(E) is rolling an even number on a die, then the complement of that event (not E)(\text{not } E) is not rolling an even number, which is the same as rolling an odd number.

The notation P(E)=pP(E) = p means that the probability of EE is the value pp. The sum of the probabilities of an event and its complement is 1.
P(E)+P(not E)=1p+P(not E)=1P(not E)=1p\begin{aligned}P(E) + P(\text{not } E) &= 1 \\ p + P(\text{not } E) &= 1 \\ P(\text{not } E) &= 1 - p\end{aligned}

Experimental Probability

Experimental probability is an estimate of how likely an event is to occur. It is based on observations of how frequently the event occurs during an experiment.
Experimental probability is an estimate of the likelihood of an event that is based on observations in an experiment. Each repetition of the experiment is called a trial. When many trials are performed, the experimental probability of an event is the ratio of the number of times the event occurs to the number of trials:
Experimental probability=Number of times the event occursTotal number of trials\text{Experimental probability} = \frac{\text{Number of times the event occurs}}{\text{Total number of trials}}
Step-By-Step Example
Estimate Experimental Probability
A parking attendant observed eight trucks, five sedans, and seven SUVs enter a parking garage. Determine the experimental probability of each type of vehicle entering the garage.
Step 1
Find the total number of vehicles that entered the garage.
8+5+7=208+5+7=20
Step 2
Identify the experimental probability for trucks.
P(truck)=820=25\begin{aligned}P(\text{truck})&=\frac{8}{20}\\&=\frac{2}{5}\end{aligned}
Step 3
Identify the experimental probability for sedans.
P(sedan)=520=14\begin{aligned}P(\text{sedan})&=\frac{5}{20}\\&=\frac{1}{4}\end{aligned}
Step 4
Identify the experimental probability for SUVs.
P(SUV)=720P(\text{SUV})=\frac{7}{20}
Solution
The experimental probabilities are:
P(truck)=25P(\text{truck})=\frac{2}{5}
P(sedan)=14P(\text{sedan})=\frac{1}{4}
P(SUV)=720P(\text{SUV})=\frac{7}{20}

Theoretical Probability

Theoretical probability is a calculation of how likely an event is to occur. It is based on a theoretical model, such as assuming that all outcomes are equally likely.
Theoretical probability is a likelihood of an event that is determined by using a model. One common model is to assume that all outcomes are equally likely. Then the theoretical probability of an event is the ratio of the number of ways the event can occur to the total number of equally likely outcomes:
Theoretical probability=Number of ways event can occurTotal number of equally likely outcomes\text{Theoretical probability} = \frac{\text{Number of ways event can occur}}{\text{Total number of equally likely outcomes}}
An experiment with equally likely outcomes is said to be fair. When using a fair die, the probability of rolling a 4 is:
P(4)=16P(4)=\frac{1}{6}
Out of six equally likely outcomes in the sample space, there is exactly one way to roll a 4.
Step-By-Step Example
Identifying the Probability with a Tree Diagram
Identify the theoretical probability of flipping exactly three heads when flipping a coin four times.
Step 1
Use a tree diagram to count the number of ways to get exactly three heads and the number of outcomes in the sample space.
There are four ways to get exactly three heads. There are 16 equally likely outcomes in the sample space.
Step 2
Calculate the ratio of the number of ways the event can occur to the number of equally likely outcomes.
P(E)=416=14P(E) = \frac{4}{16}=\frac{1}{4}
Solution
When flipping a coin four times, the probability of getting exactly three heads is 14\frac{1}{4}, or 25%.

An experiment that involves rolling a standard die was repeated 50 times. With a fair die, the theoretical probability of each outcome is 16\frac{1}{6}. However, in the experiment, the probability of rolling a one is 625\frac{6}{25}. This does not necessarily mean that the die is not fair. Probabilities found when performing experiments may not match theoretical probabilities, especially when the number of trials is small. However, as the number of trials increases, the experimental probability should approach the theoretical probability.
Result of Roll Frequency Experimental Probability
11 1212 1250=625\frac{12}{50}=\frac{6}{25}
22 77 750\frac{7}{50}
33 99 950\frac{9}{50}
44 55 550=110\frac{5}{50}=\frac{1}{10}
55 99 950\frac{9}{50}
66 88 850=425\frac{8}{50}=\frac{4}{25}

Step-By-Step Example
Compare Experimental and Theoretical Probabilities
A spinner has five equal-size sections. The table shows the results of spinning the arrow on the spinner 40 times. Determine and compare the experimental and theoretical probabilities for each color.
Result of Spin Frequency
Red 8
Blue 18
Yellow 14
Step 1
Determine the experimental probabilities by writing a ratio of the frequency (number of times the event occurs) to the total number of trials.
Result of Spin Frequency Experimental Probability
Red 88 840\frac{8}{40}
Blue 1818 1840\frac{18}{40}
Yellow 1414 1440\frac{14}{40}
Step 2
Determine the theoretical probabilities by writing a ratio of the number of ways each event can occur to the number of possible outcomes. There are 5 spaces on the spinner: 1 red, 2 blue, and 2 yellow.
Result of Spin Theoretical Probability
Red 15\frac{1}{5}
Blue 25\frac{2}{5}
Yellow 25\frac{2}{5}
Solution
Compare the experimental and theoretical probabilities. Writing them as percentages may make them easier to compare.
Result of Spin Frequency Experimental Probability Theoretical Probability
Red 88 840=20%\frac{8}{40}=20\% 15=20%\frac{1}{5}=20\%
Blue 1818 1840=45%\frac{18}{40}=45\% 25=40%\frac{2}{5}=40\%
Yellow 1414 1440=35%\frac{14}{40}=35\% 25=40%\frac{2}{5}=40\%
The experimental and theoretical probabilities are similar. In the experiment, the probability of landing on blue was slightly higher than the theoretical probability, and the probability of landing on yellow was slightly lower than the theoretical probability.