Probability – Terminology, Values & Single Events Mathematics Probability – Terminology, Values & Single Events

Lesson Objectives The aim of this powerpoint is to help you… recollect probability terms understand about the range of probability values calculate simple probabilities

What is Probability? Give YOUR suggestions before we look at mine! Did you mention the word ‘chance’ or ‘likelihood’? Probability is a value used to indicate how likely something is to happen.

Probability Terms What words do YOU know that are linked with probability? Make your suggestions… Did you get any of these…? Impossible (it will NEVER happen) Unlikely Evens (it is equally likely to happen as not happen) Likely Certain (it will happen EVERY TIME) Did you come up with any other words?

Probability Range What is the probability value of something which is IMPOSSIBLE? Make your suggestions… What is the probability value of something which is CERTAIN? Make your suggestions… All probability values fall between these two values. They should be written either as a fraction or a decimal (or possibly a percentage) but NEVER as a ratio!!! 1

Probability Range - Checks Where would EVENS come on the scale? What is its probability value? What probability value is the red arrow pointing to? How would you describe something at this point? What probability value is the blue arrow pointing to? 1 Very unlikely to happen ½ or 0.5 Likely about 1/10 or 0.1 about 7/10 or 0.7

Small Group Task Please get into 2 groups. Each group will be given a different mix and match task. TOGETHER, discuss and decide which statement links to which probability word and which probability value. Once complete, designate one person in your group to present your solution to the other group. The solutions will appear on the next two slides so do NOT move on until after your presentations.

Mix’n’Match – Set 1 1 day of the year is picked at random and it is a school day. A person who buys a ticket wins the UK national lottery. Chelsea play Swindon Town FC and WIN. I toss a fair coin and it lands on heads. The next car to pass the hospital is pink. The next person to walk into the corner shop has brown hair. You are in a maths lesson at this moment in time. You will be 10 years old tomorrow. There are 39 weeks x 5 = 195 school days each year out of 365 school days altogether so it is likely, and 195 out of 365 is about 0.53 Likely 0.53 There are millions of combinations so the chance that your single ticket has the winning combination is very very small so it is very unlikely at 0.00001 Very Unlikely 0.00001 Chelsea are a premiership team whilst Swindon Town are in the League 1 so it is highly likely they will win at about 0.05 Highly Likely 0.95 The coin is fair so it is evens at 0.5 Evens 0.5 There are VERY FEW pink cars so this occurrence is very unlikely at 0.05 Very Unlikely 0.05 Brown is a popular hair colour (but not quite half) as lots of people have fair or grey hair when you include all ages of people so unlikely at 0.4 Unlikely 0.4 You ARE doing maths so certain at 1 Certain 1 You are all at secondary school so you are all older than 10 so this is now impossible at 0 Impossible 0

Mix’n’Match – Set 2 I roll a fair die and get a number bigger than 1. It snows in London in June. A card is picked at random and it is NOT a picture card. A baby is born and it is a girl. We have a powercut tomorrow. Man Utd and Man City get a DRAW in their next match. There are 7 days in a week. Your great-great-great grandfather is still alive. 5 out of 6 numbers are bigger than 1 so getting one of these is highly likely at 0.83 Highly Likely 0.83 In the UK it is most likely to snow in winter. June is spring/summer so snowing at this time is very unlikely at 0.00001 Very Unlikely 0.00001 There are 12 picture cards but 40 other cards so getting one of those is more likely at about 0.77 Likely 0.77 Evens 0.5 Just about an equal chance of it being a boy as a girl so evens at 0.5 In the UK powercuts do not happen very often so this is very unlikely at 0.05 Very Unlikely 0.05 Both are near the top end of the Premier league so probably 0.3 each to win leaving a draw as unlikely at 0.4 Unlikely 0.4 Certain 1 Mon-Sun = 7 days so certain at 1 People don’t generally live past 100, so only usually a maximum of 4 generations (not 6!) alive so impossible at 0 Impossible 0

Individual Practice Please work through the worksheet called: ProbR-S3.xlsx When most of you have finished this worksheet, you will be asked to pass your worksheet on to the person on your left – you will mark each others. Before I read out the correct answer, you will be asked to volunteer what YOU think (not necessarily what is on the sheet you are marking) the answer is. Go through answers before moving on to next slide…

How do we find Probability values? What do YOU think? There are two main methods as follows… One method is to CALCULATE a theoretical value using the symmetries, features or properties of the situation Another method is to ESTIMATE a value using relative frequency (based on how many times something happens in an experiment or has happened in the past).

Simple Probability To calculate theoretical probability you can use the following formula: P(outcome) = No. of ways of getting this outcome Total no. of ways of getting all outcomes This formula automatically gives the probability as a fraction. To convert to a decimal simply divide the top value by the bottom value.

Example 1 – Tossing a Fair Coin If we toss a fair coin there is a total of two possible outcomes – either heads or tails. One side has heads on it and one side has tails on it. P(heads) = 1 side out of 2 sides = ½ P(tails) = 1 side out of 2 sides = ½ What is ½ as a decimal? (You should know this one automatically!) 1 ÷ 2 = 0.5

Example 2 – Rolling a fair die A fair die has a total of 6 possible outcomes as it has 6 faces, one face for each of the numbers 1 up to 6. One face has a 1 on it so P(1) = 1/6 One face has a 2 on it so P(2) = 1/6 and so on… What is the probability of rolling a 7? Please give your suggestions… No sides have a 7 on so P(7) = 0/6 or simply 0 (as it is impossible) What is the probability of rolling an even number? Please give your suggestions… 3 sides have even numbers on (2,4,6) so P(even) = 3/6 (or ½)

Example 3 – Beads in a Bag A bag contains 4 blue beads and 6 red beads. What is the probability of picking out a blue bead? Please give your suggestions… 4 blue beads out of (4+6 =) 10 beads altogether so… P(blue) = 4/10 or 0.4 (or 4/10 = 2/5) What is the probability of picking out a blue or a red bead? Please give your suggestions… ALL the beads are blue or red so this is certain so… P(B or R) = 10/10 = 1

Individual Practice Please work through the worksheet called: Prob-S2.xlsx When most of you have finished this worksheet, you will be asked to pass your worksheet on to the person on your left – you will mark each others. Before I read out the correct answer, you will be asked to volunteer what YOU think (not necessarily what is on the sheet you are marking) the answer is. Go through answers before moving on to next slide…

Total Probability We already know that an individual probability MUST be between 0 and 1. The total probability of all mutually exclusive outcomes can NOT be bigger than 1 and indeed, it adds up to 1. We’ll come back to the meaning of mutually exclusive in a moment, but first let’s go back to our previous examples. Tossing a coin  heads or tails  ½ + ½ = 1 Rolling a die  1 or 2 or 3 or 4 or 5 or 6  1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 = 1

Mutually Exclusive Outcomes that can NOT occur at the same time are mutually exclusive. Let’s look at a pack of cards… If we focus on the colours, these are mutually exclusive because if I pick a black card, it can not also be red at the same time! If we focus on suits, these are mutually exclusive because if I pick a club, it can not also be a spade or heart or diamond at the same time! However colours and suits together are NOT mutually exclusive because when I pick a card it could be red AND also a heart (or a diamond) at the same time!

P(Not happening) Something happening and something NOT happening are mutually exclusive. The total of their probabilities would add up to 1. Therefore we can use this to calculate the probability of something NOT happening by taking the probability that it WILL happen away from 1, i.e. P(will not) = 1 – P(will) Please note that the symbol for NOT is after the outcome. i.e. Probability of NOT getting a 3 is written as: P(3 )

Individual Practice Please work through the worksheet called: ProbN-S3.xlsx When most of you have finished this worksheet, you will be asked to pass your worksheet on to the person on your left – you will mark each others. Before I read out the correct answer, you will be asked to volunteer what YOU think (not necessarily what is on the sheet you are marking) the answer is. Go through answers before moving on to next slide…

Notes and Examples In your exercise books or on paper, please write up your own notes and examples on everything we have covered during this lesson. It may help to include answers to the following questions… What is probability? What different terms (or words) describe probability? What range of values must probability lie between and in what form must it be written and NEVER be written? How can we calculate simple probability – give a description and then write the formula? How can we estimate probability? What does ‘mutually exclusive’ mean? How do we calculate the probability of something NOT happening?

What next? Make sure you have completed the worksheets: ProbR-S3.xlsx, Prob-S2.xlsx and ProbN-S3.xlsx Make sure you have completed writing up your notes. Get your teacher to check your notes. If there is any time left, you may start the experiments that you will be doing at the start of the next lesson – ask your teacher what to do!