4.1.1 Basic probability
Probability is a measure of the likelihood or chance that a specific event or outcome will occur, expressed as a number between \( 0 \) and \( 1 \).
A probability of \( 0 \) means that the event is impossible, while a probability of \( 1 \) means that the event is certain. In between, probabilities range from very unlikely (close to \( 0 \) ) to very likely (close to \( 1 \)).
Probabilities can be estimated based on prior knowledge, experience, or data analysis.
Keywords and terminology
Event:
An event is a set of outcomes of an experiment.
For example, if you roll a die, the event of getting an even number consists of the outcomes \( 2 \), \( 4 \), and \( 6 \).
Sample space:
The sample space is the set of all possible outcomes of an experiment.
For example, if you flip a coin, the sample space consists of the outcomes “heads” and “tails”.
Probability distribution:
A probability distribution is a function that assigns probabilities to each possible outcome of an experiment.
For example, the probability distribution of rolling a die is uniform, meaning that each outcome \( (1, 2, 3, 4, 5, or 6) \) has an equal probability of \( \frac{1}{6} \).
Random variable:
A random variable is a variable that takes on different values depending on the outcome of a random event.
For example, if you roll a die, the number you roll is a random variable.
Expected value:
The expected value of a random variable is the average value it would take over an infinite number of trials.
For example, the expected value of rolling a die is \( \frac{(1+2+3+4+5+6)}{6}= 3.5 \)
Conditional probability:
Conditional probability is the probability of an event occurring given that another event has occurred.
For example, the probability of rolling a \( 3 \) given that you rolled an odd number is \( \frac{1}{3} \), because there are three odd numbers (\( 1 \), \( 3 \), and \( 5 \)) and only one of them is a \( 3 \).
Bayes’ theorem:
Bayes’ theorem is a formula for calculating conditional probabilities.
It states that the probability of an event \( A \) given an event \( B \) is equal to the probability of event \( B \) given event \( A \) times the probability of event \( A \), divided by the probability of event \( B \).
Basic formulas
The formula for calculating probability depends on the type of situation you are dealing with. Here are some common formulas:
Probability of a simple event:
For a simple event with only two possible outcomes (e.g. flipping a coin), the probability of one outcome is:
\( Probability=\frac {Number \ of \ favorable \ outcomes}{Total \ number \ of \ possible \ outcomes} \)
For example, the probability of getting a head when flipping a fair coin is \( \frac{1}{2} \), because there is one favorable outcome (getting a head) out of two possible outcomes (getting a head or getting a tail).
Probability of a compound event:
For a compound event with more than one possible outcome (e.g. rolling a die and getting an even number), the probability is:
\( Probability=\frac{Number \ of \ favorable \ outcomes}{Total \ number \ of \ possible \ outcomes} \)
For example, the probability of rolling a die and getting an even number is \( \frac{3}{6} \) or \( \frac{1}{2} \), because there are three favorable outcomes (getting a \( 2 \), \( 4 \), or \( 6 \)) out of six possible outcomes (getting a \( 1 \), \( 2 \), \( 3 \), \( 4 \), \( 5 \), or \( 6 \) ).
Probability of independent events:
If two events are independent (i.e., the outcome of one event does not affect the outcome of the other), the probability of both events occurring is:
\( Probability \ of \ both \ events=Probability \ of \ event \ A \times Probability \ of \ event \ B \)
For example, the probability of flipping a coin and getting heads twice in a row is
\( (\frac{1}{2}) \times (\frac{1}{2}) = \frac{1}{4} \)
because the probability of getting heads on the first flip is \( \frac{1}{2} \) and the probability of getting heads on the second flip is also \( \frac{1}{2} \).
Probability of dependent events:
If two events are dependent (i.e., the outcome of one event affects the outcome of the other), the probability of both events occurring is:
\( Probability \ of \ both \ events=Probability \ of \ event \ A \times Probability \ of \ event \ B \ given \ that \ event \ A \ has \ occurred \)
For example, the probability of drawing two cards from a deck without replacement and getting two aces is
\( (\frac{4}{52}) \times (\frac{3}{51}) \)
because the probability of drawing an ace on the first draw is \( \frac{4}{52} \) and the probability of drawing an ace on the second draw, given that an ace was already drawn, is \( \frac{3}{51} \).
Missing probabilities
To find missing probabilities, you need to use the information you have and apply the laws of probability. Here are some steps to follow:
Identify what is being asked and what information is given.
This could involve identifying the event and the probability that is missing.
Depending on the problem, you may need to use one or more of the following laws of probability:
The probability of the union of two events is the sum of their individual probabilities minus the probability of their intersection.
The probability of the intersection of two independent events is the product of their individual probabilities.
The probability of an event \( A \) given that another event \( B \) has occurred is equal to the probability of the intersection of \( A \) and \( B \) divided by the probability of \( B \).
Use the information and laws of probability to solve for the missing probability.
Make sure that your answer makes sense in the context of the problem and that it is between \( 0 \) and \( 1 \).
Worked example:
The table shows the probabilities that a biased dice will land on \( 2 \), on \( 3 \), on \( 4 \), on \( 5 \) and on \( 6 \).
Neymar rolls the biased dice \( 200 \) times.
Work out an estimate for the total number of times the dice will land on \( 1 \) or on \( 3 \).
Use the fact that the total of all probabilities must sum to \( 1 \).
Let’s say the missing probability is \( x \).
\( x+0.17+0.18+0.09+0.15+0.1=1 \)
\( x+0.69=1 \)
\( x=1-0.69 \)
\( x=0.31 \)
The probability of landing on \( 1 \) or on \( 3 \) is the sum of the probability of landing on \( 1 \) and the probability of landing on \( 3 \).
To estimate the total number of times the dice will land on \( 1 \) or on \( 3 \), multiply it by the number of times the dice rolled
\( 200 \times (0.31+0.18)=98 \)