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I teach mutually exclusive outcomes directly after students have encountered Venn diagrams. This is the fifth lesson in the Year 8 Probability, Outcomes and Venn diagrams scheme of work.

Before progressing on to mutually exclusive outcomes students review Venn diagrams using the starter question below.

Students present their solutions to me on mini whiteboards so I can check their understanding and address any common misconceptions. It is important students have a good understanding of the intersection of sets A and B as this is necessary for the remainder of the lesson.

Understanding mutual exclusivity as two or more sets with no intersection is a natural extension of Venn diagrams. The starter question showed all the students were comfortable working with sets that overlap. To extend this concept I ask the class to sketch a Venn diagram for getting a Head or Tail when a fair coin is tossed. The diagram below shows the most common response.

To feedback, I ask students to consider what outcome would lie within the intersection of the two sets and re-consider their response. Nearly all students show the diagram below.

Next, I ask everyone to sketch a Venn diagram on their mini whiteboards of rolling an odd or even number on a dice. Event A are the odd numbers and Event B the even numbers. To encourage discussion students work in pairs on a single whiteboard. To promote peer support, I may ask the less able student within a pair to do the writing. This makes the more able student become the teacher.

A few minutes into the task some pairs continue to struggle linking their understanding of mutual exclusivity to Venn diagrams. To support, we discuss how the intersection between the two sets has to equal zero, or P(Odd ∩ Even) = 0. Most students present a Venn diagram like the one below.

To take this example further we discuss mutually exclusive events that cover all the possibilities are called **exhaustive** events. The probabilities of mutually exclusive events that are also exhaustive add up to 1.

Using the example of it raining or not raining I ask the
class to show me the probability of it not raining if there is a 30% chance of
rain. All students currently show me
P(no rain) = 70%. I share with the
students the formula *P(Event not happening) = 1 – P(Event happening)*.

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After we have worked through the examples on the second slide students attempt the questions on the third slide independently. The differentiated worksheet provides further consolidation and extension problems.

The plenary challenges students to explain why two events are mutually exclusive and to use this fact to calculate the probability of an event.

This typically takes about 8 minutes with students free to work on their own or with the person next to them. Although all everyone agrees winning or losing are mutually exclusive some of the class are unable to sufficiently explain why. I encourage other students to explain their reasoning. The lack of border between the two regions being the key.

Most students were able to correctly work out the probability of Stephen winning. All but three students considered the whole circle. The most able realised win or lose equals 60°. Therefore, winning equals 20/60 = 1/3.

My name is Jonathan Robinson and I am passionate about teaching mathematics. I am currently Head of Maths in the South East of England and have been teaching for over 15 years. I am proud to have helped teachers all over the world to continue to engage and inspire their students with my lessons.

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