Mathematics is known for its certainty. If we add $2$ and $2$, we’re always going to get $4$. If we multiply $3$ and $4$ we always get $12$. These things are certain and they will never change. However, in real life, sometimes we have situations that are not certain. Their possible outcome changes each time we repeat the action. A popular way of making a decision is flipping a coin. If we flip a coin, we don’t know if we’re going to get heads or tails. If we play a board game and roll a die, we can’t be certain which number we’re going to get.
The branch of mathematics that specifies in events where we have no guaranteed certainty is called probability.
Probability of event is a value that tells us how likely something is to happen. Instead of “probability” of an event we can also use “odds” or “chances” of an event, which mean the same thing.
Example. Flipping a coin
Let’s go back to our example of flipping a coin. Each time we flip a coin, we get either heads or tails. Those are $2$ possible outcomes and each one is just as likely. If we flip a coin a lot of times, it’s reasonable to expect that half the flips will be heads, and the other half would be tails.
As a result, the probability of flipping heads is $\frac{1}{2}$. But the probability of flipping tails is also $\frac{1}{2}$.
Probability line
For better understanding of probability, we’ll introduce a probability line.
The probability of an event occurring is somewhere between impossible and certain. Translated to mathematics’ language, $0$ represents an impossible event, and $1$ represents certain event. An event can’t be less likely than impossible, and more likely than certain.
In addition to writing probability as fractions, we can also write them as decimals or percentages since it’s easy to convert between them.
For example, when flipping a coin, the probability of getting tails is equal to $\frac{1}{2}$ or $0.5$ or $50\%$. We would say that event has “even chance of happening and not happening“, or the event is “just as likely to happen as it is not to happen“.
Unlikely event has probability between $0$ and $\frac{1}{2}$ or less than $50\%$. Likely event has probability greater than $50\%$ but less than $100\%$. Probability of certain event would be $1$ or $100\%$.
However, it is important to remember that probability doesn’t tell us what will happen for certain, just what will happen on average. It means that the more trials you do, the closer you get to expected probability. For example, if we flip a coin $10$, it doesn’t mean $50\%$ of it will always be heads and $50\%$ will be tails. In fact, results may be way off, we could get $2$ heads and $8$ tails. However, the more we repeat the process the closer the outcome will get to its expected probability.
Example. Rolling a die
When rolling a die, there are $6$ values we can get, ${1, 2, 3, 4, 5, 6}$. Those are $6$ possible outcomes and they are all equally likely to happen. It is certain we’ll get one of these numbers, but since we have $6$ possibilities, we have to divide the value of certainty among all the possibilities. We have to divide $1$ since it represents certain event by $6$. In conclusion, the probability of each event equals $\frac{1}{6}$.
Important
As we saw above, the odds of flipping heads is $\frac{1}{2}$ and the odds of flipping tails is $\frac{1}{2}$. If we add up those probabilities, we get $1$.
Or the odds of rolling number $6$ on a die is $\frac{1}{6}$ and the chances of rolling any other number is also $\displaystyle{\frac{1}{6}}$. If we add up all the probabilities, we get $1$ once again. $$\displaystyle{\frac{1}{6} + \frac{1}{6} +\frac{1}{6} +\frac{1}{6} +\frac{1}{6} +\frac{1}{6} = \frac{6}{6} = 1}$$
That’s not a coincidence. If we add up probabilities of all possible outcomes of an event we get $1$ or $100\%$. That’s because it is certain at least one of the possible outcomes will happen, i.e. the event will occur.
Definition of probability
These two examples are enough to see a pattern. In general, we calculate probability by dividing the number of ways it can happen by the total number of possible outcomes.
The probability of an event $A$ is written as $P(A)$.
Example. There are $7$ marbles in a bag. $4$ are red, $2$ are green, and $1$ is blue. What are the odds of picking green?
Let $A$ be event: we picked green marble.
Number of ways we can pick green marble is $2$, since we have $2$ outcomes that give us what we want. We can pick one or the other, it’s not important which. Total number of possible outcomes is $7$ since we have $7$ marbles in a bag and every single one can be picked.
By using the formula above, we have $$\displaystyle{P(A)=\frac{2}{7}}$$
Example. When rolling a die, what are the odds of getting an even number?
$A$ is event: we get an even number.
Number of ways we can get an even number is $3$ since $\{2, 4, 6\}$ are the only even numbers on a die. Total number of possible outcomes is $6$ since there are $6$ numbers on a die.
Let’s calculate the possibility:$$\displaystyle{P(A)=\frac{3}{6} = \frac{1}{2}}$$
Example. When rolling $2$ dice, what are the odds of getting two numbers whose sum is equal to $7$ or $11$.
$A$ is event: sum of numbers rolled on dice is $7$ or $11$.
Since this is the way we defined $A$ we can explicitly show all possible outcomes for $A$ to happen, $A=\{{(1,6), (6,1), (2,5),(5,2),(3,4),(4,3),(5,6),(6,5)}\}$. The number of ways we can get two numbers whose sum is equal to $7$ or $11$ is $8$. When we had only one die, total number of possible outcome was $6$. Now that we have two, that number is equal to $6\cdot 6=36$. Finally, the possibility of $A$ is $$\displaystyle{P(A)=\frac{8}{36} = \frac{2}{9}}$$
Complement
When we flip a coin, if the event is heads, its complement is tails. When we roll a die, if the event is getting even number, it’s complement is getting an odd number. So the complement of event is all outcomes other than the ones we want, meaning they are mutually exclusive. Complement of an event $A$ is often written as $A^{c}$. Event and its complement together make all possible outcomes.
Since probability of $A$ is written as $P(A)$, the notation of probability of its complement is $P(A^{c})$.
As we said earlier, the probability of all possible events must be equal $1$. Since event and its complement make all possible outcomes, we have $$P(A)+P(A^{c})=1$$
Sometimes it’s easier to calculate the probability of a complement, than the actual event. Well see that in next example.
Example. Let’s roll two dice. What are the odds of getting different numbers?
Let $A$ be our wanted outcome. If we wanted to write the whole list, it would be quite a long one. $$A=\{(1,2), (1,3), (1,4)…(2,1),(2,3)…(6,5)\}.$$ However, the complement of $A$ would be “both numbers on dice are equal”. What would $A^{c}$ look like? $$A^{c}=\{(1,1),(2,2),(3,3),(4,4),(5,5),(6,6)\}.$$ As we can see, $A^{c}$ has only $6$ outcomes, which is a lot less than the number of outcomes for $A$. Now lets calculate the probability of $A^{c}$. $$\displaystyle{P(A^{c})=\frac{6}{36}=\frac{1}{6}}.$$ Then the probability of $A$ is equal to
$$\displaystyle{P(A)=1-P(A^{c})=1-\frac{1}{6}=\frac{5}{6}}.$$
Example. A single card is chosen from a standard deck of $52$ playing cards. What is the probability of choosing a card that is not a king?
There are two possible ways of calculating this probability. We could calculate the probability of getting every other card or we could calculate the probability of getting a king and then finding the probability of its complement. Lets try the latter.
Lets say $A$ is event “we chose a king”. Then $A^{c}$ would be “we didn’t choose a king”.
A standard deck of $52$ cards has $4$ kings, so the probability of $A$ is $$\displaystyle{P(A)=\frac{4}{52}}$$
Now we can calculate the probability of $A^{c}$ using the formula above.
$$P(A)+P(A^{c})=1$$ $$P(A^{c})=1-P(A)$$ $$P(A^{c})=1-\displaystyle{\frac{4}{52}}$$
Which gives us $$P(A^{c})=\displaystyle{\frac{48}{52} = \frac{12}{13}}$$
