GCSE Maths Practice: conditional-probability

Question 7 of 13

This question shows that conditional probability can sometimes equal 1 when all remaining outcomes match.

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\( \begin{array}{l}\text{A box contains 6 red balls and 4 blue balls.} \\ \text{One ball is drawn at random.} \\ \text{What is the probability it is red, given that it is not blue?}\end{array} \)

Choose one option:

If only one type of outcome remains after applying the condition, the probability is 1.

When Conditional Probability Equals 1

In probability, a value of 1 represents absolute certainty. This question is designed to show that conditional probability can sometimes result in a probability of exactly 1 when all remaining outcomes satisfy the required condition.

The key phrase in the question is "given that it is not blue". This immediately removes all blue balls from consideration. Once those outcomes are excluded, only red balls remain. Since every possible remaining outcome is red, the probability of selecting a red ball becomes 1.

Understanding Why the Probability Is 1

Probability is calculated using:

Number of favourable outcomes ÷ total number of possible outcomes

After applying the condition, both the number of favourable outcomes and the total number of outcomes are the same. When these values are equal, the probability is always 1.

This does not mean the event was guaranteed before the condition was applied. It becomes certain only because additional information has restricted the sample space.

Step-by-Step Method

  1. Identify the condition given in the question.
  2. Remove all outcomes that do not meet the condition.
  3. Check how many different outcomes remain.
  4. If all remaining outcomes match the desired result, the probability is 1.

Worked Example 1

A bag contains 7 apples and 3 oranges. One fruit is chosen. What is the probability the fruit is an apple, given that it is not an orange?

Answer: Removing oranges leaves only apples. The probability is 1.

Worked Example 2

A drawer contains 5 black socks and 5 white socks. One sock is taken. What is the probability the sock is black, given that it is not white?

Answer: Removing white socks leaves only black socks. The probability is 1.

Common Mistakes

  • Thinking probability cannot be equal to 1.
  • Using the original total instead of the restricted total.
  • Assuming there must always be uncertainty.
  • Second-guessing a correct answer because it looks “too simple”.

Real-Life Meaning

Conditional certainty appears often in real life. For example, if you know a day is a weekday, the probability it is Saturday is 0. If you know it is not Saturday or Sunday, the probability it is a weekday is 1. New information can turn uncertainty into certainty.

Frequently Asked Questions

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

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

Can probability ever be greater than 1?
No. All probabilities must be between 0 and 1 inclusive.

Study Tip

If every remaining outcome satisfies the condition, do not overthink it. The probability is 1.

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