2The following topics will be covered The Complement of an eventCombined eventsMutually exclusive eventsExhaustive eventsConditional ProbabilityIndependent eventsUsing tree diagrams
3½ Remember: The probability of an event happening = number of wanted outcomestotal number of outcomesEarlier, we placed the following events on the probability line.324156Impossible1CertainGetting a tail when tossing a coin.Getting a 3 when you throw a die.Being born in the month of March.213Choosing a red cube at random from the bag.46Choosing a blue bead at random from the bag.5Getting a 5 on the spinner.
4½ Remember: The probability of an event happening = number of wanted outcomestotal number of outcomesDiscuss the probabilities of the following events and their placement on the probability line.651423Impossible1CertainNot choosing a blue bead at random from the bag.Not getting a tail when tossing a coin.Not getting a 3 when you throw a die.Not being born in the month of March.23Not choosing a red cube at random from the bag.465Not getting a 5 on the spinner.11125671214112
5From what we have done below it should be clear that: The probability of an event not happening = 1 - probability of it happeningIf we call the event “A” then symbolically we have:P(not A) = 1 - P(A)Impossible1CertainNot choosing a blue bead at random from the bag.Not getting a tail when tossing a coin.Not getting a 3 when you throw a die.Not being born in the month of March.23Not choosing a red cube at random from the bag.465Not getting a 5 on the spinner.11127
6The probability of an event not happening = 1 - probability of it happening P(not A) = 1 - P(A)(c)(a)(b)(f)(e)(d)1Jenny has 12 cards with different shapes on as shown. She turns the cards over and chooses one at random. Mark the probabilities for the chosen card on your number line.(a) P(red shape)(b) P(not red shape)(c) P(3D shape)(d) P(not 3D shape)(e) P(not triangular shape)(f) P(not quadrilateral shape)5/127/122/1210/129/128/12
7The probability of an event not happening = 1 - probability of it happening P(not A) = 1 - P(A)(d)(c)(a)(e)1(f)(b)Sam has 6 cards with different numbers on as shown. He turns the cards over and chooses one at random. Mark the probabilities for the chosen card on your number line.31724(a) P(not even)(b) P(not prime)(c) P(not a multiple of 3)(d) P(not less than 18)(e) P(not greater than 20)(f) P(not less than 7 factors)4/63/6157122/61/65/61/6
8The complement of an event A’ The complement of an event contains all the outcomes not contained in that eventP(A’) = 1 – P(A)The probability of an event not happening = 1 - probability of it happeningP(not A) = 1 - P(A)
10Combined eventsA card is selected at random from a deck of cards. Find the probability that the card is aA) kingB) a heartC) king of heartsD) either a king or heart
11Do I want a king and a heart? The probability of a king or a heart does not include the card being a king and a heart so to calculate this probability we mustP (K U H) = 4/ /52 – 1/52Generally we can say thatP(A U B) = P(A) + P(B) – P(A n B)We do not want to double count the area of intersection!
12Another exampleThese are the number of students who study Maths, Biology, both or neither35122726
14For Combined Events P(A U B) = P(A) + P(B) – P(A n B) Don’t count the intersection twice!
15Mutually exclusiveThrowing a coin and tossing a die have outcomes that are not the sameEvents that have no common outcomes are called mutually exclusive.The intersection is zero as they have no common elements!
16Mutually exclusive When two events are mutually exclusive then: P(A U B) = P(A) + P(B)
17ExampleGiven that events A and B are mutually exclusive where P(A) = 3/10 andP(B) = 2/5 find the value of P(A U B)
18What is the difference between? For starters, there is a link between mutually exclusive events- they can't both happen at once. However, there is no link between independent events- they don't effect each other at all. It might be easier to understand if you also consider non-mutually-exclusive events and dependent events.If I draw one card from a deck, drawing an ace and drawing a king are mutually exclusive events- a single card cannot be both an ace and a king. However, drawing an ace and drawing a spade are not mutually exclusive events- a single card can be both an ace and a spade.If I draw one card, return that card to the deck, and then draw another card, the draws are independent of each other- the sample space is the same for both draws because I returned the first card to the deck. If I draw one card, but do not return that card to the deck, and then draw another card, these events are dependent- the sample space is different since I didn't return the first card to the deck. Say I drew an ace the first time. Then there is one less card and one less ace in the deck, so the probabilities for the second draw have changed. So mutually exclusive events are contrasted with non-mutually-exclusive events, asking whether one event excludes the other. Independent events are contrasted with dependent events, asking whether one event effects the probability of the other.
19Exhaustive eventsIf two events together contain all the possible outcomes then we say the two events are exhaustive.ExampleEvent A = headEvent B = tailBoth events A and B together are exhaustive as you can only get a head/ tail on a coin (all possible outcomes)So P(A) + P(B) = ??
20Exhaustive events P (A U B) = 1 There are no elements outside the two events
21ExampleP (A U B) = 1Given that P(A) = 4/5 and P(B) = ½ and P(A n B) = 3/10 show that A and B are exhaustive.
22Some more examplesGiven that P(A) = 0.55, P(AUB) = 0.7 and P(A n B) = 0.2 find P(B’)
23Another exampleGiven P(G’) = 5x, P(H) = 3/5 P(GUH) = 8x and P(G n H) = 3x find the value of x.
24Conditional Probability If you are given two events but told that one of the events has already occurred then this is called conditional probability.This is because if an events has already occurred it will influence what is going to happen in the future.The probability of an event A given B =P(A I B) = P(AnB)P(B)
25ExampleP(A I B) = P(AnB)P(B)Two fair dice are thrown. Find the probability that one of the dice shows a four given that the total of the dice is ten. P( four I ten).Firstly write down all the possible outcomes for the two dice. (page 374)Four and a ten means (4,6) and (6,4)P( four U ten) = 2/36P( ten) = 3/36 (using 4,6 5,5 and 6,4)So P (four I ten) = 2/3
26Example 2 Given P(A) = ½ P(AIB) = ¼ and P(AUB)= 2/3 find P(B). P(A I B) = P(AnB)P(B)Given P(A) = ½ P(AIB) = ¼ and P(AUB)= 2/3 find P(B).
27For independent events A and B, P(A n B) = P(A) x P(B) IndependenceP(A I B) = P(AnB)P(B)Two events that have no effect on each are called independent eventsP(AIB) = P(A) (the event B doesn’t influence A!)P(A n B) = P(A)P(B)The throwing of 2 dice are independent events. This means that the outcomes on one die are not affected in any way by the outcomes on the other.For independent events A and B, P(A n B) = P(A) x P(B)
28ExampleA card is picked at random and a fair die is thrown. Find the probability that the card is the Ace of hearts and the die shows a 6.