GCSE Maths Practice: conditional-probability

Question 10 of 13

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

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\( \begin{array}{l}\text{A group of 50 students contains 20 males and 30 females.} \\ \text{If a student is selected at random, what is the probability the student is female,} \\ \text{given that they are not male?}\end{array} \)

Choose one option:

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

Conditional Probability When the Outcome Is Certain

This question demonstrates a special but important case of conditional probability: situations where the condition removes all uncertainty. When every remaining outcome satisfies the event being asked about, the conditional probability becomes exactly 1.

The condition "given that the student is not male" immediately removes all male students from consideration. Once this restriction is applied, the sample space consists only of female students. Because there are no other possibilities left, the probability of selecting a female student becomes certain.

How Conditional Probability Changes the Sample Space

Originally, the group contains both males and females. However, conditional probability does not always use the original group. Instead, it uses a restricted sample space based on the information given.

In this case, the condition removes all males. The new sample space contains only females, so every possible outcome satisfies the event.

Step-by-Step Reasoning

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

Worked Example 1

A bag contains only cats and dogs. If an animal is chosen and is known not to be a dog, what is the probability it is a cat?

Answer: Removing dogs leaves only cats, so the probability is 1.

Worked Example 2

A set of numbers contains only even and odd numbers. If a number is chosen and is known not to be odd, what is the probability it is even?

Answer: Once odd numbers are excluded, only even numbers remain, so the probability is 1.

Common Mistakes

  • Thinking probability must always involve fractions.
  • Using the original total instead of the restricted total.
  • Overthinking a logically certain situation.
  • Assuming probability cannot be equal to 1.

Real-Life Interpretation

Conditional certainty appears often in everyday reasoning. If you know a person is not a child, and the only other category is adult, then the probability the person is an adult is 1. New information can eliminate uncertainty entirely.

Frequently Asked Questions

Is a probability of 1 valid?
Yes. It means the event is guaranteed under the given condition.

Does this mean the event was guaranteed originally?
No. The certainty comes from the condition, not the original situation.

Can probability ever exceed 1?
No. Probabilities always lie between 0 and 1 inclusive.

Study Tip

If the condition logically forces the outcome to occur, trust your reasoning. A probability of 1 is correct.

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