Probability Ch 14 IB standard Level

The following topics will be covered The Complement of an event Combined events Mutually exclusive events Exhaustive events Conditional Probability Independent events Using tree diagrams

½ Remember: The probability of an event happening = number of wanted outcomes total number of outcomes Earlier, we placed the following events on the probability line. 3 2 4 1 5 6 Impossible 1 Certain ½ Getting a tail when tossing a coin. Getting a 3 when you throw a die. Being born in the month of March. 2 1 3 Choosing a red cube at random from the bag. 4 6 Choosing a blue bead at random from the bag. 5 Getting a 5 on the spinner.

½ Remember: The probability of an event happening = number of wanted outcomes total number of outcomes Discuss the probabilities of the following events and their placement on the probability line. 6 5 1 4 2 3 Impossible 1 Certain ½ Not 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. 2 3 Not choosing a red cube at random from the bag. 4 6 5 Not getting a 5 on the spinner. 11 12 5 6 7 12 1 4 1 12

From what we have done below it should be clear that: The probability of an event not happening = 1 - probability of it happening If we call the event “A” then symbolically we have: P(not A) = 1 - P(A) Impossible 1 Certain ½ Not 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. 2 3 Not choosing a red cube at random from the bag. 4 6 5 Not getting a 5 on the spinner. 11 12 7

The probability of an event not happening = 1 - probability of it happening P(not A) = 1 - P(A) (c) (a) (b) (f) (e) (d) ½ 1 Jenny 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) 5/12 7/12 2/12 10/12 9/12 8/12

The probability of an event not happening = 1 - probability of it happening P(not A) = 1 - P(A) (d) (c) (a) (e) ½ 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. 3 17 24 (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) 4/6 3/6 15 7 12 2/6 1/6 5/6 1/6

The complement of an event A’ The complement of an event contains all the outcomes not contained in that event P(A’) = 1 – P(A) The probability of an event not happening = 1 - probability of it happening P(not A) = 1 - P(A)

A deck of cards

Combined events A card is selected at random from a deck of cards. Find the probability that the card is a A) king B) a heart C) king of hearts D) either a king or heart

Do 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 must P (K U H) = 4/52 + 13/52 – 1/52 Generally we can say that P(A U B) = P(A) + P(B) – P(A n B) We do not want to double count the area of intersection!

Another example These are the number of students who study Maths, Biology, both or neither 35 12 27 26

P(M) = P(B) = P(MnB) = P(B') = (not French) P(MuB) = P(MnB') = 35 12 27 26 n = intersect u = union F E P(M) = P(B) = P(MnB) = P(B') = (not French) P(MuB) = P(MnB') = P(AuB) = P(A) + P(B) - P(AnB) The ADDITION rule

For Combined Events P(A U B) = P(A) + P(B) – P(A n B) Don’t count the intersection twice!

Mutually exclusive Throwing a coin and tossing a die have outcomes that are not the same Events that have no common outcomes are called mutually exclusive. The intersection is zero as they have no common elements!

Mutually exclusive When two events are mutually exclusive then: P(A U B) = P(A) + P(B)

Example Given that events A and B are mutually exclusive where P(A) = 3/10 and P(B) = 2/5 find the value of P(A U B)

What 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.

Exhaustive events If two events together contain all the possible outcomes then we say the two events are exhaustive. Example Event A = head Event B = tail Both 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) = ??

Exhaustive events P (A U B) = 1 There are no elements outside the two events

Example P (A U B) = 1 Given that P(A) = 4/5 and P(B) = ½ and P(A n B) = 3/10 show that A and B are exhaustive.

Some more examples Given that P(A) = 0.55, P(AUB) = 0.7 and P(A n B) = 0.2 find P(B’)

Another example Given P(G’) = 5x, P(H) = 3/5 P(GUH) = 8x and P(G n H) = 3x find the value of x.

Conditional 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)

Example P(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) = 2/36 P( ten) = 3/36 (using 4,6 5,5 and 6,4) So P (four I ten) = 2/3

Example 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).

For independent events A and B, P(A n B) = P(A) x P(B) Independence P(A I B) = P(AnB) P(B) Two events that have no effect on each are called independent events P(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)

Example A 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.