# Theoretical Probability

Students are required to be able to calculate theoretical probability in a variety of different situations. The activities in this resource list provide opportunity for students to generate theoretical possibility spaces for single and combined events with equally likely and mutually exclusive outcomes. Students use these results to calculate theoretical probabilities and show that they know that the probabilities of an exhaustive set of mutually exclusive outcomes add up to one.

Visit the secondary mathematics webpage to access all lists.

### Shopping Mall

This video resource raises quick-fire starter questions. A question is posed for each topic, offering a pause point for teachers to hold a freeze frame on screen while students discuss the question. The starter appropriate to this topic is:

**Probability: Assorted Chocolates**

Students are given the number of different types of chocolates in a bag and asked to identify the probability of a single event and combining two events. The relevant section of the video begins at 09:13.

This flash game expects students to use their knowledge of probability to choose whether the next card in the sequence is more likely to be higher or lower than the one being displayed.

The number of cards to guess in a game can be varied and there is also the option of playing with a single suit or a full deck of cards. The game allows students to look at probabilities based upon those cards which have already been revealed, but this will cost points.

This activity could be used as a starter to lead into the principles of probabilty and as a plenary to test understanding.

### Probability

This resource contains investigations, worksheets and practical activities. In pack two the activities appropriate for this topic is **What's the probability?** in which students find simple probabilities.

In pack three, **Probability **requires students to find probabilities using playing cards, dice and coins in single and combined events. **Combined probability** is a practical introduction to using tree diagrams.

### Probability

These resources contain worked examples to support learning and are followed by focused exercises for use by students to practise skills. Activities to reinforce learning and offer extension opportunities are also supplied, as are topic related tests.

Probability part A covers simple probability, the outcome of two events and finding probabilities using relative frequency. The task continues by asking students to determine probabilities, find the probability of an outcome from two events, use tree diagrams and use multiplication for independent events and mutually exclusive events.

In part B students explore misconceptions, evens and odds, tossing three coins and throwing two dice.

### Too Many Boys in the Family?

A question posed by this resource from CensusAtSchool, investigating whether probability relates to reality. Real data is contrasted with the theoretical probabilities of the number of boys in a family, using the assumption that the probability of having a boy is 0.5. Equally likely outcomes, combined events, tree diagrams, binomial distribution, theoretical and experimental probability are all explored.

### Probability

This resource begins with work requiring students to show their knowledge of using the probability scale before moving on to find the probability of a single event, the probability of two events, the use of tree diagrams and a comparison of theoretical and experimental probabilities. Each section contains worked examples followed by an exercise.

### Winning the Lottery

Aimed at students in Key Stage Four, this task from CensusAtSchool uses the context of the National Lottery to look at theoretical probabilities and compare these to a experimental data. The worksheet contains a link to a site to simulate a lottery draw. The activity allows students to investigate successive events and use tree diagrams.

Students will gain experience of working with equally likely outcomes, experimental probability and relative frequency.

### Three Dice

In this activity students play a bingo-style game. To maximize their chances of winning, students must decide which numbers are most likely to occur when three dice are thrown and the scores are added. This activity starts with a game. In order to maximize their chances of winning the game, pupils must decide which numbers are most likely to occur when three dice are thrown and the scores are added together. The resource includes pupil stimulus, teacher guidance, progression table and an interactive worksheet. You will need to check compatibility of the interactive part of the activity.

### Rabbits

In this activity students learn that a simulation is a model of a real situation. Data for the birth rate of rabbits is provided and is used by students to develop their own model to simulate the breeding of rabbits in a field and consider the likely damage to the farmer’s crop. Initially this is done as a recording process, using an existing example and template, which requires students to process information provided. It offers a good opportunity for group collaboration.

An excel spreadsheet allows students to extend the activity and investigate more complex scenarios. They are able to alter the number of years, maximum litter size and the initial number of females of breeding age, as well as adjusting gender bias and the maximum possible monthly death rate. The resource includes a complete set of student resources as well as notes for teachers.

This investigation is based on the probability of balls, which are dropped through a triangular array of pins (a quincunx), into slots beneath. Every time a ball hits a nail it has a probability of fifty percent to fall to the left of the nail and a probability of fifty percent to fall to the right of the nail. As balls accumulate in the slots beneath the triangle they will resemble a binomial distribution.

This model allows students to vary the number of layers in the Quincunx, the speed of the drop and the weighting given to the left- right movement of the balls. Use it to predict the outcomes of future experiments and that the experimental results tend to a theoretical distributions as the number of experiments increase.