GCSE Maths Practice: conditional-probability

Question 9 of 13

This question shows how a condition can remove all uncertainty in a probability problem.

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\( \begin{array}{l}\text{A group of 30 students includes 18 girls and 12 boys.} \\ \text{If a student is chosen at random, what is the probability that the student is a girl,} \\ \text{given that the student is not a boy?}\end{array} \)

Choose one option:

If the condition guarantees the outcome, the probability is 1.

Conditional Probability and Logical Certainty

This question demonstrates an important idea in conditional probability: sometimes the given condition removes all uncertainty. When this happens, the probability of the remaining outcome becomes exactly 1.

The key phrase here is "given that the student is not a boy". This condition immediately removes all boys from the sample space. Once this information is applied, the only students who can possibly be chosen are girls.

Why the Probability Becomes 1

Probability is always calculated as:

Number of favourable outcomes ÷ total number of possible outcomes

After applying the condition, the total number of possible outcomes is the same as the number of favourable outcomes. Every student who remains in the sample space is a girl. When this happens, the probability is equal to 1, meaning the event is certain.

This certainty does not come from the original group. It comes from the additional information provided by the condition.

Step-by-Step Reasoning

  1. Identify the condition in the question.
  2. Translate the condition into plain language.
  3. Remove all outcomes that do not satisfy the condition.
  4. Check what outcomes remain.
  5. If all remaining outcomes match the event, the probability is 1.

Worked Example 1

A box contains only apples and oranges. If a fruit is chosen and is known not to be an orange, what is the probability that it is an apple?

Answer: Once oranges are excluded, only apples remain, so the probability is 1.

Worked Example 2

A class has students who study either French or Spanish. If a student is selected and is known not to study Spanish, what is the probability the student studies French?

Answer: Removing Spanish students leaves only French students, so the probability is 1.

Common Misconceptions

  • Thinking probability can never be equal to 1.
  • Trying to calculate fractions when logical reasoning is enough.
  • Using the original group instead of the restricted group.
  • Overthinking a situation that is logically certain.

Real-Life Interpretation

This type of conditional certainty appears frequently in real life. For example, if you know a day is not Saturday or Sunday, then the probability it is a weekday is 1. Additional information can turn uncertainty into certainty.

Frequently Asked Questions

Is this really a probability question?
Yes. It shows how probability and logic work together when conditions are applied.

Can probability ever be greater than 1?
No. Probability values are always between 0 and 1 inclusive.

Why include questions like this?
They build confidence and prevent misconceptions about probability.

Study Tip

If the condition logically guarantees the outcome, trust your reasoning. A probability of 1 is valid and correct.

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