Probability

The probability of an event occurring is between 0 and 1 If an event is certain not to happen, the probability is 0 eg: the probability of getting a 7 when you roll a die = 0 If an event is sure to happen, the probability is 1 eg: the probability of getting either a head or a tail when you flip a coin = 1 All other events have a probability between 0 and 1

Likely or unlikely? Not likely Likely Impossible to happento happen Certain Very unlikelyEqual chanceVery Likely to happenof happening to happen

Relative Frequency This gives information about how often an event occurred compared with other events. eg: Maths Exam results from 26 students Exam %No. of studentsRelative Frequency > < 401 = 0.38 (2 dp) = 0.46 (2dp) = 0.12 (2 dp) = 0.04 (2 dp)

Sample Space The set of all possible outcomes is called the sample space. eg. If a die is rolled the sample space is: { 1, 2, 3, 4, 5, 6 } eg. If a coin is flipped the sample space is: { H, T } eg. For a 2 child family the sample space is: { BB, BG, GB, GG }

Equally likely outcomes In many situations we can assume outcomes are equally likely. When events are equally likely: Equally likely outcomes may come from, for example: experiments with coins, dice, spinners and packs of cards Probability = Number of favourable outcomes Number of possible outcomes

Pr (getting a 5 when rolling a dice) = Favourable Outcomes are the results we want Possible Outcomes are all the results that are possible - the SAMPLE SPACE Examples: Pr (even number on a dice) = Pr (J, Q, K or Ace in a pack of cards) = =

Lattice Diagrams A spinner with numbers 1 to 4 is spun and an unbiased coin is tossed. Graph the sample space and use it to give the probabilities: (a) P(head and a 4)(b) P(head or a 4) Sample space P(head and a 4) = P(head or a 4) = (those shaded)

Sample Space is {H1, H2, H3, H4, T1, T2, T3, T4} A spinner with numbers 1 to 4 is spun and an unbiased coin is tossed. We can work out the sample space by a lattice diagram or a tree diagram. Lattice Diagram

Lattice Diagrams The sample space when rolling 2 dice can be shown by the following lattice diagram: (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) Pr (double) == Pr (total ≥ 7) = =

Tree Diagrams for Probability A tree diagram is a useful way to work out probabilities. B B B B B B B G G G G G G G eg: Show the possible combination of genders in a 3 child family 1 st child 2 nd child 3 rd child Outcomes BBB BBG BGB BGG GBB GBG GGB GGG Pr (2 girls & a boy) =   

Experimental Probability When we estimate a probability based on an experiment, we call the probability by the term “relative frequency”. Relative frequency = The larger the number of trials, the closer the experimental probability (relative frequency) is to the theoretical probability. Probability Relative Frequency Theoretical Experimental Relating factor “is the term we use in”

Pr (getting a 5 when rolling a dice) = Favourable Outcomes are the results we want Possible Outcomes are all the results that are possible – (the sample space) Examples: Pr (even number on a dice) = Pr (J, Q, K or Ace in a pack of cards) == Complementary Events When rolling a die, ‘getting a 5’ and ‘not getting a 5’ are complementary events. Their probabilities add up to 1. Pr (getting a 5) =Pr (not getting a 5) =

Using Grids (Lattice Diagrams) to find Probabilities red die blue die ● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Pr (double) = = Pr (total ≥ 7) = = coin die H T ● ● ● ● ● ●● ● ● ● ● ● Rolling 2 dice: Rolling a die & Flipping a coin: Pr (tail and a 5) = Pr (tail or a 5) =

Using Grids (Lattice Diagrams) to find Probabilities (1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6) (3,1) (3,2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4) (4,5) (4,6) (5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6) Pr (double) == Pr (total ≥ 7) = =

Multiplying Probabilities In the previous lattice diagram, when rolling a die and flipping a coin, coin die H T ● ● ● ● ● ●● ● ● ● ● ● Pr (tail) =Pr (5) = Pr (tail and a 5) = x = So Pr (A and B) = Pr (A) x Pr (B) example:Jo has probability ¾ of hitting a target, and Ann has probability of ⅓ of hitting a target. If they both fire simultaneously at the target, what is the probability that: a) they both hit itb) they both miss it ie Pr (Jo hits and Ann hits) = Pr (Jo hits) x Pr (Ann hits) = ¾ x ⅓ = ¼ ie Pr (Jo misses and Ann misses) ie Pr (Jo misses) x Pr (Ann misses) = ¼ x ⅔ =

Tree Diagrams to find Probabilities In the above example about Jo and Ann hitting targets, we can work out the probabilities using a tree diagram. Let H = hit, and M = miss Jo’s results H M ¾ ¼ Ann’s results H M H M ⅓ ⅔ ⅓ ⅔ Outcomes H and H H and M M and H M and M Probability ¾ x ⅓ = ¼ ¾ x ⅔ = ½ ¼ x ⅓ = ¼ x ⅔ = total = 1 Pr (both hit) = ¼ Pr (both miss) = Pr (only one hits) ie Pr (Jo or Ann hits) = ½ + = ● ● or

Expectation When flipping a coin the probability of getting a head is ½, therefore if we flip the coin 100 times we expect to get a head 50 times. Expected Number = probability of an event occurring x the number of trials eg: Each time a rugby player kicks for goal he has a ¾ chance of being successful. If, in a particular game, he has 12 kicks for goal, how many goals would you expect him to kick? Solution: Pr (goal) = ¾ Number of trials = 12 Expected number = ¾ x 12 = 9 goals