Calculating Probabilities Statistics and Probability

The language of probability Probability is a measurement of the chance or likelihood of an event happening. Words that we might use to describe probabilities include: unlikely chance likely possible probable certain poor chance impossible very likely even chance

If a dice has ten faces, then there are ten possible outcomes, one for each face of the dice. An event can have several outcomes. Outcomes and events What are the possible outcomes when throwing a ten-sided dice? Can you think of an event that has two outcomes? A simple example of an event that has two outcomes is flipping a coin. The two outcomes are heads and tails.

Outcomes and events Each outcome of a given event has a probability or a chance of occurring. What are the chances of each outcome from throwing a ten-sided dice? Assuming that the dice is fair, the chances of each outcome occurring is. 1 10

The probability scale The chance of an event happening can be shown on a probability scale. impossiblecertaineven chanceunlikely likely Less likelyMore likely Meeting with King Henry VIII A day of the week starting with a T The next baby born being a boy Getting homework this lesson A square having four right angles

The probability scale We measure probability on a scale from 0 to 1. If an event is impossible or has no chance of occurring, then it has a probability of 0. If an event is certain it has a probability of 1. This can be shown on the probability scale as: Probabilities are written as fractions, decimals and, less often, as percentages. impossiblecertaineven chance

The probability scale

Higher or lower

Listing possible outcomes When you roll a fair dice you are equally likely to get one of six possible outcomes: Since each number on the dice is equally likely the probability of getting any one of the numbers is 1 divided by 6 or. 1 6

Calculating probability What is the probability of the following events? P(tails) = 1 2 P(red) = 1 4 P(7 of ) = 1 52 P(Friday) = 1 7 2) This spinner stopping on the red section? 3) Drawing a seven of hearts from a pack of 52 cards? 4) A baby being born on a Friday? 1) A coin landing tails up?

Calculating probability If the outcomes of an event are equally likely then we can calculate the probability using the formula: Probability of an event = Number of successful outcomes Total number of possible outcomes For example, a bag contains 1 yellow, 3 green, 4 blue and 2 red marbles. What is the probability of pulling a green marble from the bag without looking? P(green) = 3 10 or 0.3 or 30%

Using the probability scale

Calculating probability This spinner has 8 equal divisions: a)a red sector? b)a blue sector? c)a green sector? What is the probability of the spinner landing on a) P(red) = 2 8 = 1 4 b) P(blue) = 1 8 c) P(green) = 4 8 = 1 2

Calculating probabilities with spinners

Calculating probability A fair dice is thrown. What is the probability of getting a)a 2? b)a multiple of 3? c)an odd number? d)a prime number? e)a number bigger than 6? f)an integer? a) P(2) = 1 6 b) P(a multiple of 3) = 2 6 = 1 3 c) P(an odd number) = 3 6 = 1 2

Calculating probability A fair dice is thrown. What is the probability of getting a)a 2? b)a multiple of 3? c)an odd number? d)a prime number? e)a number bigger than 6? f)an integer? d) P(a prime number) = 3 6 e) P(a number bigger than 6) = f) P(an integer) = 6 6 = 1 = Don’t write 0 6

Calculating probability The children in a class were asked how many siblings (brothers and sisters) they had. The results are shown in this frequency table: Number of siblings Number of pupils What is the probability that a pupil chosen at random from the class will have two siblings? There are 30 pupils in the class and 9 of them have two siblings. So, P(two siblings) = 9 30 = 3 10

Calculating probability A bag contains 12 blue balls and some red balls. The probability of drawing a blue ball at random from the bag is. How many red balls are there in the bag? balls represent of the total. 3 7 So, 4 balls represent of the total 1 7 and, 28 balls represent of the total. 7 7 The number of red balls = 28 – 12 =16

The probability of an event not occurring If the probability of an event occurring is p then the probability of it not occurring is 1 – p. The following spinner is spun once: What is the probability of it landing on the yellow sector? P(yellow) = 1 4 What is the probability of it not landing on the yellow sector? P(not yellow) = 3 4

The probability of an event not occurring The probability of a factory component being faulty is What is the probability of a randomly chosen component not being faulty? P(not faulty) = 1 – 0.03 =0.97 The probability of pulling a picture card out of a full deck of cards is. What is the probability not pulling out a picture card? 3 13 P(not a picture card) = 1 – =

The probability of an event not occurring The following table shows the probabilities of 4 events. For each one work out the probability of the event not occurring. Event Probability of the event occurring Probability of the event not occurring ABCD % %

The probability of an event not occurring There are 60 candies in a bag. What is the probability that a candy chosen at random from the bag is: a) Not a cola bottle 5 6 P(not a cola bottle) = b) Not a bear P(not a bear) = 10 are cola bottles, 1 4 are fried eggs,the rest are bears.20 are hearts, = 3 4

Experimental Probability The probability of landing on a 6 when a dice is thrown is 1/6. Experimental Probability = You can estimate the probability by doing an experiment. Number of Successful Trials Total Number of Trials This is called the calculated probability