Probability & Two-Way Tables — Core Skills

Probability & Two-Way Tables: The Basics

Part 1 of 2 — Wanted Over Total

Probability is a number that says how likely something is. You write it as a fraction, and the fraction is always built the same way:

$\text{probability} = \frac{\text{number of outcomes you want}}{\text{total number of outcomes}}$

The top is what you are hoping for. The bottom is everything that could happen. Nothing else changes.

Worked example: a bag of marbles

A bag holds $3$ red marbles, $5$ blue marbles, and $2$ green marbles. One marble is drawn at random. What is the probability it is red?

Step 1 — Find the total. Add every marble: $3 + 5 = 8$, then $8 + 2 = 10$. There are $10$ marbles in all.

Step 2 — Find how many you want. There are $3$ red marbles.

Step 3 — Write the fraction. The probability of drawing red is $\frac{3}{10}$.

The word "not"

If a question asks for the probability of not red, count everything that is not red. That is the $5$ blue plus the $2$ green, which is $5 + 2 = 7$ marbles. So the probability of drawing a marble that is not red is $\frac{7}{10}$.

When the question hands you a smaller group

Sometimes a question narrows the group down before it asks. Here is a two-way table of $40$ students:

	Yes	No	Total
Juniors	15	5	20
Seniors	9	11	20
Total	24	16	40

A junior is picked at random. What is the probability that this student said Yes?

The words "a junior is picked" tell you the group is only the juniors. So the bottom of the fraction is $20$, not $40$. The top is the $15$ juniors who said Yes.

$\text{probability} = \frac{15}{20}$

Both numbers divide by $5$: $15 \div 5 = 3$ and $20 \div 5 = 4$, so the answer is $\frac{3}{4}$.

Read the first few words of the question carefully. They tell you what goes on the bottom.

Probability & Two-Way Tables: Practice

Part 2 of 2 — Three Steps, Every Time

1. Decide what group the question is about. If it says "a student is picked at random," the group is everyone. If it says "a junior is picked," the group is only the juniors. That group total goes on the bottom.
2. Count how many in that group match what is being asked. That count goes on the top.
3. Write the fraction and reduce it if you can. Divide the top and the bottom by the same number until you cannot go any further.

Reducing, in one line

$\frac{12}{30}$: both numbers divide by $6$, and $12 \div 6 = 2$ while $30 \div 6 = 5$, so $\frac{12}{30} = \frac{2}{5}$.

The word "not"

For a "not" question, count everything that does not match, and put that count on top. In a bag of $10$ marbles where $3$ are red, "not red" is $10 - 3 = 7$ marbles, so the probability is $\frac{7}{10}$.

Working from a table

If the number you need is not printed in the table, add the cells to get it. A row added across gives that row's total. A column added down gives that column's total. The corner of the table is everyone.

Every probability answer is between $0$ and $1$. If your fraction comes out bigger than $1$, the top and bottom got swapped.