This pdf covers the following topics:- Complementary Events Addition Rule Disjoint Events Non-Disjoint Events
1. Complementary Events Addition rule
2. Complementary Events the events of one outcome happening and that outcomes not happening are complementary (opposite) ( not E is contrary event to E ) E Not E For example : you pick up a card from a deck E: P(Heart)= ¼ Not E: P(not Heart)= ¾
3. Complementary Events THE SUM of the of the probabilities of complementary events is 1. Not E from which I get: : Probability of the contrary event Not E is: “1 minus the Probability of the event E” Not E
4. Complementary Events You pick up a card from deck of 52 cards. Which is the probability of picking a figures? 12 P( figures) = 52 Which is the probability of not picking a figures ? 12 52 − 12 40 P(NotFigures) = 1− = = 52 52 52
5. Complementary Events 1) The probability that it will rain tomorrow is 0.4 . What is the probability it does not rain ? P(not Rain) = 1-0.4 = 0.6 2) Tossing 2 coins,which is the probability of: a) never getting Tail ? P(never T) = P(Head Head) = 1/4 b) getting at least once Tail? (TT or HT or TH ) P(at Least Once T)=1-P(never T)=1-1/4= 3/4
6. ADDITION RULE PROBABILITY OF A OR B
7. 1) Disjoint Events There are two situations 2) NOT Disjoint Events Disjoint Events ? Two events are Disjoint ( Mutually Exclusive ) if they can't happen at the same time Turning left and turning right are Mutually Exclusive (you can't do both at the same time) Cards: Kings and Aces are disjoint What is Not Disjoint ( not Mutually Exclusive ) ? Turning left and scratching your head can happen at the same time Cards: Kings and Hearts, because we can have a King of Hearts!
8. 1) DISJOINT Events (Mutually Exclusive ) A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing an Ace or a King? P(ACE or KING ) = P(Ace) + P(King) = 4/52 + 4/52 = 8/52
9. 1) DISJOINT events Addition Rule B A for DISJOINT Events: When two events A and B are disjoint, the probability that A or B will occur is: the SUM of the Probability of each Event. P(A or B) = P(A) + P(B)
10. 2) NOT DISJOINT ( NOT Mutually Exclusive) example A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing an Heart or a King? P(H or K) = P(H) + P(K) - P(both) =13/52 + 4/52 –1/52 = 16/52
11. A and B = intersection 1) NOT DISJOINT ADDITION RULE for NOT disjoint A or B = union Events When two events A and B are NOT DISJOINT, the probability that A or B will occur is : the SUM of the probability of each event, MINUS the probability of the overlap. both P(A or B) = P(A) + P(B) - P(A and B) U union ∩ Intersection
12. SUMMARY : ADDITION RULE DISJOINT EVENTS NOT DISJOINT EVENTS Mutually Exclusive Not Mutually Exclusive A and B together is impossible: P(A and B) = 0 A and B together is possible ! P(A or B) = P(A) + P(B) P(A or B) = P(A) + P(B) − P(A and B)
13. TRY IT YOURSELF TEST 1: A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing an Ace or a figure? 2: A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing an Ace or Red Card? 3: You are going to roll two dice. Find: P(sum that is even or sum that is a multiple of 3).
14. 1: A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing an Ace or a figure? Figures P(Ace)=4/52 Aces These events are mutually exclusive ( disjoint) since they cannot occur at the same time. P(A or B) = P(A) + P(B) U union P(Ace OR Figure) = 4/52+12/52 = 16/52
15. 2. A single card is chosen at random from a standard deck of 52 playing cards. What is the probability of choosing an Ace or Red Card? Aces Red cards P(Red card)=26/52 These events are NOT disjoint since they have some overlap ( favorable outcomes in common ) P(A or B) = P(A) + P(B) - P(A and B) U union ∩ Intersection P(Ace OR Red Card) = 4/52+26/52-2/52 = 28/52
16. ANSWER 3 3. You are going to roll two dice. Find P(sum that is even or sum that is a multiple of 3). The addition rule says we need to find P(even) + P(multiple of 3) - P(both) The number of possible outcomes of rolling two dice = 36 P(even) means how many ways to roll:2, 4, 6, 8, 10, or 12. P(even) = 18/36 P(multiple of 3) means how many ways to roll : 3, 6, 9 or 12. P(multiple of 3) = 12/36 P(both) means what is the overlap. Notice that 6 and 12 occur in both places and have been counted twice. We need to subtract those out. P(both) = 6/36 P(even or multiple of 3)= 18/36 + 12/36 - 6/36 = 24/36
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