Probability calculations: Year 10: Planning tool

Expected level of development

Australian Curriculum Mathematics V9: AC9M10P01

Numeracy Progression: Understanding chance: P6

At this level, students extend their knowledge and skills with the introduction of conditional probability. For the first time, students consider events that might depend on a preceding event. The probability of some outcomes shifts when additional information is accounted for.

By now, students are familiar with the probability of dice and cards. They know that the probability of drawing a queen from a full deck of cards at random is 452. Relying on those odds to make the perfect hand seems risky. But what if it was known that the next card had a face on it? Then the probability of completing the hand would increase significantly to 412 , which are much more favourable odds. Adding this layer of sophistication allows for probabilistic scenarios that more closely reflect the real world, and can lead to interesting class discussion by introducing the language of probability as it intersects with students’ lives. For example, 'If you want to increase your chances of being a successful mathematics learner, then you could…’ or ‘What events that have already happened, and might be out of your control, have contributed to your chances of success as a mathematics learner?’

Explicitly teach the language of conditional probability, as it is nuanced, and students will need to use and interpret terms such as ‘if … then’, ‘given’, ‘of’ and ‘knowing that’ interchangeably. There is also a key formula that must be understood and implemented: Pr(A|B)= Pr(A∩B) Pr(B) . This formula can be intuited, and students should be encouraged to think about why that specific ratio produces the conditional probability of A and B. Use worked examples to illustrate how the formula can be rearranged to provide alternative results. Tree diagrams are an effective tool in this instance, as the branches represent conditional probabilities. Students will be familiar with using tree diagrams for repeated or consecutive events, such as picking marbles out of a bag. Demonstrate how they might be used in a different way, such as calculating a batter’s probability of being dismissed against a bowling attack with both pace and spin bowlers. What are the chances that the batter will face a spin/pace bowler? How does their chance of being dismissed vary between pace and spin?

Students should almost exclusively be required to analyse and interpret information presented in worded scenarios. They will need to practise applying probability notation and assigning probabilities to this information. Ensure that students can identify which event depends on the other so that they can correctly label them as A and B. The formula cannot be effectively applied if students can't make this distinction. Venn diagrams are also useful for visualising these problems. Encourage students to use their hands to cover up the unnecessary area of the diagram. If B has already happened, then students only need to consider the elements contained in B (including the intersection), and can ignore those that are only in A. This process can elicit a deeper understanding of the formula, as the two values present in the ratio are those that are now visible in the Venn diagram.

Teaching and learning summary:

Introduce the language and formula of conditional probability.
Ensure students can identify and interpret statements of conditional probability.
Allow students to consolidate understanding by complementing interpretation of conditional probability with tree diagrams, Venn diagrams and two-way tables.
Ensure students can describe their approach and results.

Students:

are familiar with conditional probability statements
can interpret conditional scenarios
use diagrams to aid understanding and assign probabilities
can describe conditional scenarios and their results.

Some students may:

misinterpret the Pr(A|B) is the probability of B given A, rather than A given B. Have students separate the statements as separate phrases, and consider which event depends on which.
be confused in worded questions where it is difficult to identify that the analysis is one of conditional probability. Have students use Venn diagrams as part of the breakdown of the question, and initially use worded statements rather than A and B.
struggle to adapt when a question asks students to determine the probability of the intersection, or of B. Students need to identify which probabilities they are given and which they need to find.

The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.

Learning intention

I am learning to understand how one event defines the condition for another event to occur.

Why are we learning about this?

Learning about conditional probability will ground you with a skill that can be used in the world around you, from predicting and making decisions in the field of health and medicine to financial contexts used by governments and large private enterprises.

What to do

This game is designed so that you can master the skill of identifying conditional probability scenarios by experimenting with home objects.

Find two standard dice or borrow them from a friend or school.
One by one, roll both dice.
What is the probability that you roll a 4 on both dice?
Use a tree diagram to help you. What is the probability that your second roll is under 4 if the first roll is under 6?
What is the probability that your second roll is not a 4?
What is the probability that your first roll is not a 4 if your second roll is not a 6?

Success criteria

I can conduct a simple experiment with an object from home to better understand conditional probability.

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