Basic probability

Probability is the chance of an event happening, written as a fraction. The number of favourable outcomes (the ones you want) goes on top, and the total number of possible outcomes goes on the bottom.

Suppose a box holds $6$ white balls and $4$ black balls, $10$ in total, and you draw one without looking.
What is the probability it is black?
The favourable outcomes are the $4$ black balls, out of $10$ total, so the probability is $\frac{4}{10} = \frac{2}{5}$.

When more than one outcome counts as a success, add their counts.
Suppose a bag holds $5$ red, $3$ green, and $2$ blue counters, $10$ in total.
For the probability of drawing red or green, add those: $5 + 3 = 8$ favourable, out of $10$, so $\frac{8}{10} = \frac{4}{5}$.

Often the counts come from a two-way table. The table below sorts $200$ students by whether they play a sport and whether they play an instrument.

To find the probability that a randomly chosen student plays a sport, take the sport total over the grand total:
$\frac{100}{200} = \frac{1}{2}$.

Sometimes the question restricts you to one group. If you are told a student plays no sport, the total is only the $100$ students with no sport.
Of those, the table shows $80$ play an instrument, so the probability is $\frac{80}{100} = \frac{4}{5}$.

A bag holds $3$ red and $5$ blue marbles, and you draw one at random.
What is the probability of drawing a red marble?
Favourable outcomes $= 3$ (red), total outcomes $= 3 + 5 = 8$.
So the probability is $\frac{3}{8}$.

A jar holds $4$ red, $3$ green, and $5$ blue marbles, and you draw one at random.
What is the probability of drawing a red or a green marble?
Add the favourable counts: $4 + 3 = 7$, out of $12$ total.
So the probability is $\frac{7}{12}$.

Using the table above, a student is known to play an instrument.
What is the probability they play no sport?
Limit to the $120$ instrument players; of these, $80$ play no sport.
So the probability is $\frac{80}{120} = \frac{2}{3}$.