Basic Probability
By the end of this lesson, you’ll be able to:
- Interpret probability as a measure of how likely an event is to occur.
- Identify outcomes, events, and sample spaces.
- Calculate simple probabilities using equally likely outcomes.
- Find probabilities from tables and observed data.
- Use the complement rule to find the probability that an event does not occur.
- Distinguish between theoretical and experimental probability.
Key Ideas
Probability measures how likely an event is to occur.
A probability always falls between:
\[ \boxed{0 \le P(A) \le 1} \]
where:
- \(0\) means the event is impossible
- \(1\) means the event is certain
- values closer to \(1\) mean the event is more likely
- values closer to \(0\) mean the event is less likely
For example:
\[ P(A)=0.8 \]
represents an event that is more likely to occur than an event with:
\[ P(B)=0.2 \]
Outcomes, Sample Spaces, and Events
An outcome is one possible result of an experiment.
A sample space is the set of all possible outcomes.
An event is one outcome or a group of outcomes that we are interested in.
For example, when rolling a standard six-sided die, the sample space is:
\[ \{1,2,3,4,5,6\} \]
If the event is “roll an even number,” the favorable outcomes are:
\[ \{2,4,6\} \]
There are 3 favorable outcomes out of 6 possible outcomes.
Basic Probability Formula
When all outcomes are equally likely:
\[ \boxed{ P(\text{event}) = \frac{\text{number of favorable outcomes}} {\text{total number of possible outcomes}} } \]
For the die example:
\[ P(\text{even}) = \frac{3}{6} = \boxed{\frac{1}{2}} \]
Probability can be written as a fraction, decimal, or percent:
\[ \frac{1}{2}=0.5=50\% \]
Complementary Events
The complement of an event \(A\) is the event that \(A\) does not occur.
Because either \(A\) occurs or it does not:
\[ P(A)+P(\text{not }A)=1 \]
Therefore:
\[ \boxed{ P(\text{not }A)=1-P(A) } \]
For example, suppose:
\[ P(\text{rain})=0.30 \]
Then:
\[ P(\text{no rain}) = 1-0.30 = \boxed{0.70} \]
Probability From a Table
Probability problems often give information using a frequency table.
Suppose 25 students are asked to choose their favorite ice-cream flavor.
| Category | Count |
|---|---|
| Vanilla | 12 |
| Chocolate | 8 |
| Strawberry | 5 |
| Total | 25 |
To find the probability that a randomly selected student prefers chocolate, use:
\[ P(\text{chocolate}) = \frac{\text{students who prefer chocolate}} {\text{total students}} \]
So:
\[ P(\text{chocolate}) = \frac{8}{25} = \boxed{\frac{8}{25}} \]
As a decimal:
\[ \frac{8}{25}=0.32 \]
or:
\[ 32\% \]
The same idea works for any category in the table.
For example:
\[ P(\text{vanilla})=\frac{12}{25} \]
and:
\[ P(\text{strawberry}) = \frac{5}{25} = \boxed{\frac{1}{5}} \]
Theoretical vs. Experimental Probability
There are two important ways probability can be determined.
Theoretical Probability
Theoretical probability is based on what should happen according to the possible outcomes.
For a fair six-sided die:
\[ P(\text{rolling a 4}) = \boxed{\frac{1}{6}} \]
because exactly one of the six equally likely outcomes is a 4.
Experimental Probability
Experimental probability is based on what actually happened in observed trials or collected data.
Use:
\[ \boxed{ P(\text{event}) = \frac{\text{number of times the event occurred}} {\text{total number of trials}} } \]
Suppose a coin is flipped 100 times and lands on heads 47 times.
The experimental probability of heads is:
\[ P(\text{heads}) = \frac{47}{100} = \boxed{0.47} \]
This does not have to equal the theoretical probability exactly.
For a fair coin, the theoretical probability is:
\[ \frac{1}{2}=0.50 \]
but an experiment may produce a value such as \(0.47\), \(0.52\), or another nearby value.
Theoretical probability describes what we expect based on the probability model.
Experimental probability describes what was actually observed.
With many repeated trials, experimental probability often gets closer to the theoretical probability.
Common Problem Types
1. Basic Probability From Equally Likely Outcomes
Count the favorable outcomes and divide by the total number of possible outcomes.
Use:
\[ P(\text{event})=\frac{\text{favorable outcomes}}{\text{total outcomes}} \]
2. Probability of Multiple Favorable Outcomes
An event can contain more than one favorable outcome.
Count all outcomes that satisfy the condition.
3. Probability From a Table or List
Use the frequencies given in the problem.
The relevant count becomes the numerator, and the total count becomes the denominator.
4. Complement Probability
Sometimes it is easier to calculate the probability that an event does not occur.
Use:
\[ P(\text{not }A) = 1-P(A) \]
5. Basic Probability With Cards
A standard deck contains:
\[ 52 \]
cards.
There are:
- 4 suits: hearts, diamonds, clubs, spades
- 13 cards in each suit
- 26 red cards
- 26 black cards
- 12 face cards: 4 jacks, 4 queens, and 4 kings
6. Experimental Probability
Use the observed number of successes and total number of trials.
\[ P(\text{event})=\frac{\text{observed successes}}{\text{total trials}} \]
7. Finding a Missing Count From Probability
You may be given a probability and the total number of observations.
Multiply the probability by the total number of observations.
Strategies
- Identify exactly what event the question is asking about.
- Determine the total number of possible outcomes.
- Count the favorable outcomes.
- When outcomes are equally likely, use:
\[ P(A) = \frac{\text{favorable outcomes}} {\text{total outcomes}} \]
- Reduce fractions when possible.
- Convert between fractions, decimals, and percents when needed.
- Use the complement rule for questions involving not.
- If the outcomes are difficult to visualize, write out the sample space.
- For experimental probability, use observed frequencies rather than what theoretically should happen.
- Check that your final probability is between 0 and 1.
Worked Examples
Example 1 — Coin Flip
A fair coin is flipped.
What is the probability of heads?
The sample space is:
\[ \{\text{heads},\text{tails}\} \]
There is 1 favorable outcome out of 2 equally likely outcomes.
Therefore:
\[ P(\text{heads}) = \boxed{\frac{1}{2}} \]
Example 2 — Probability From a Bag
A bag contains:
- 3 red marbles
- 5 blue marbles
- 2 green marbles
First find the total:
\[ 3+5+2=10 \]
There are 5 favorable outcomes for blue.
Therefore:
\[ P(\text{blue}) = \frac{5}{10} = \boxed{\frac{1}{2}} \]
Example 3 — Probability With Cards
A card is drawn randomly from a standard 52-card deck.
What is the probability of drawing a spade?
There are:
\[ 13 \]
spades and:
\[ 52 \]
total cards.
Therefore:
\[ P(\text{spade}) = \frac{13}{52} = \boxed{\frac{1}{4}} \]
Example 4 — Multiple Favorable Outcomes
A standard die is rolled.
What is the probability of rolling a number less than 5?
The possible outcomes are:
\[ \{1,2,3,4,5,6\} \]
The favorable outcomes are:
\[ \{1,2,3,4\} \]
So:
\[ P(\text{less than }5) = \frac{4}{6} = \boxed{\frac{2}{3}} \]
Example 5 — Complement
The probability that a randomly selected student rides the bus to school is:
\[ 0.65 \]
What is the probability that the student does not ride the bus?
Use the complement rule:
\[ P(\text{not bus}) = 1-P(\text{bus}) \]
Substitute:
\[ P(\text{not bus}) = 1-0.65 = \boxed{0.35} \]
Example 6 — Experimental Probability
A basketball player makes 42 of 60 free throws.
Based on these results, what is the experimental probability that the player makes a free throw?
Use:
\[ P(\text{make}) = \frac{\text{successful trials}} {\text{total trials}} \]
So:
\[ P(\text{make}) = \frac{42}{60} = \frac{7}{10} = \boxed{0.70} \]
- Dividing by the number of favorable outcomes instead of the total number of outcomes.
- Using favorable ÷ total when the possible outcomes are not equally likely.
- Forgetting to include all possible outcomes in the sample space.
- Giving a probability less than 0 or greater than 1.
- Confusing theoretical probability with experimental probability.
- Forgetting that \(P(\text{not }A)=1-P(A)\).
- Forgetting to simplify fractions when appropriate.
- Assuming a random experiment must produce exactly the theoretical proportions in a small number of trials.
Practice Problems
A fair six-sided die is rolled. What is the probability of rolling a 2?
A jar contains 4 red marbles and 6 yellow marbles. What is the probability of randomly selecting a yellow marble?
A standard card is drawn from a 52-card deck. What is the probability that it is a face card?
A standard die is rolled. What is the probability of rolling an even number?
If 9 out of 30 surveyed students walk to school, what is the experimental probability that a randomly selected surveyed student walks to school?
The probability that an event occurs is \(0.72\). What is the probability that it does not occur?
A spinner has 8 equal sections numbered 1 through 8. What is the probability of landing on a number greater than 5?
1. A six-sided die has six equally likely outcomes:
\[ \{1,2,3,4,5,6\} \]
Only one outcome is a 2.
Therefore:
\[ \boxed{P(2)=\frac{1}{6}} \]
2. First find the total number of marbles:
\[ 4+6=10 \]
There are 6 yellow marbles.
Therefore:
\[ P(\text{yellow}) = \frac{6}{10} = \boxed{\frac{3}{5}} \]
3. A standard deck has 3 face cards in each of 4 suits:
\[ 3(4)=12 \]
There are 52 total cards.
Therefore:
\[ P(\text{face card}) = \frac{12}{52} = \boxed{\frac{3}{13}} \]
4. The even numbers on a standard die are:
\[ \{2,4,6\} \]
There are 3 favorable outcomes out of 6 total outcomes.
Therefore:
\[ P(\text{even}) = \frac{3}{6} = \boxed{\frac{1}{2}} \]
5. This probability is based on observed survey data, so use experimental probability:
\[ P(\text{walk}) = \frac{9}{30} = \boxed{\frac{3}{10}} \]
As a decimal:
\[ P(\text{walk})=\boxed{0.30} \]
6. Use the complement rule:
\[ P(\text{not }A) = 1-P(A) \]
Substitute:
\[ P(\text{not }A) = 1-0.72 = \boxed{0.28} \]
7. The possible outcomes are:
\[ \{1,2,3,4,5,6,7,8\} \]
Numbers greater than 5 are:
\[ \{6,7,8\} \]
There are 3 favorable outcomes out of 8 total outcomes.
Therefore:
\[ \boxed{ P(\text{greater than }5)=\frac{3}{8} } \]
Summary
Probability measures how likely an event is to occur.
For equally likely outcomes:
\[ \boxed{ P(\text{event}) = \frac{\text{favorable outcomes}} {\text{total outcomes}} } \]
Every probability satisfies:
\[ \boxed{ 0\le P(A)\le1 } \]
The complement rule is:
\[ \boxed{ P(\text{not }A)=1-P(A) } \]
Theoretical probability is based on the possible outcomes of a probability model.
Experimental probability is based on observed results.
- Identify the event first.
- Count favorable outcomes.
- Count total outcomes.
- Make sure outcomes are equally likely before using favorable ÷ total.
- Probability must be between 0 and 1.
- \(0\) = impossible; \(1\) = certain.
- For not, think complement: \(1-P(A)\).
- Experimental probability comes from actual observed data.
- Write the sample space when the possible outcomes are unclear.