Probability and Conditional Probability

Write the eligible group before the fraction

In a class of 30 students, 18 play a sport. Selecting a student at random gives a sport-player probability of $\frac{18}{30}=\frac{3}{5}$. The denominator counts everyone who could be selected.

If the question restricts the selection to a subgroup, that full-class denominator no longer applies. Rewrite the sample space before using the table's counts.

Read conditional probability within a row or column

Suppose 12 students take art, and 8 of those 12 also play a sport. Given that a selected student takes art, the probability that the student plays a sport is $\frac{8}{12}=\frac{2}{3}$.

The numerator is the intersection of the two groups. The denominator is the group after "given that": art students. Reversing the condition would use the total number of sport players instead.

Use complements and avoid double counting

If a probability is 0.35, the probability of its complement is $1-0.35=0.65$. For a union, add the counts in the two groups and subtract their overlap so those observations count once.

If 18 students play a sport, 12 take art, and 8 do both, then 22 do at least one: $18+12-8=22$. The remaining 8 of the 30 do neither.

Multiply only when the conditions justify it

For two independent coin flips, the chance of two heads is $\frac{1}{2}\cdot \frac{1}{2}=\frac{1}{4}$. For drawing two red counters without replacement, the second probability changes after the first draw.

From 3 red counters among 5 total, the chance of two reds without replacement is $\frac{3}{5}\cdot \frac{2}{4}=\frac{3}{10}$. Write the second denominator after the first item is removed.

Check your understanding

Sample probability and conditional probability questions

Practice this skill

1. State the sample space in words before making the fraction.
2. Practice both directions of the same conditional question.
3. Check whether repeated selections replace the first item.

Print the probability and conditional probability worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.