Pencil red pen highlighter GP notebook textbook calculator
Pencil, red pen, highlighter, GP notebook, textbook, calculator Answer the following. Similar types of problems where the most missed on the last test. 1. Factor completely across the rationals. a) b) +3 2. Solve for all solutions of x. a) +2 +3 b) total:
2. Solve for all solutions of x. a) b) +1 +1 +1 +2 total: +2
Part 1: Let’s formalize AND and OR: compound events – any event combining _____ 2 or ____ more ________ simple events. exclusive or _______? mutually inclusive Consider: are the events mutually _______ Mutually Exclusive (or ______) CANNOT occur disjoint – events that _______________ overlap at the same time. That is, events that do not ____. Examples: • Drawing a 5 or drawing a King from a standard deck of cards • Rolling a pair of dice to get a sum of 5 or a sum of 7 P(A) + P(B) Formula: P (A or B) = _______ Practice: Given a standard deck of cards, find each probability. 1. P (even number card or king) = 2. P (ace or queen) =
Mutually Inclusive – events that can occur at the _____. same time ________ Examples: • Drawing a card that is both a 5 and a heart from a standard deck of cards. • Rolling a pair of dice to get a 6 and rolling an even number Since inclusive events can happen simultaneously, DO NOT double count the probability when the events ____. overlap ______ Formula: P (A or B) = P(A) + P(B) – P(A and B) P (A and B) = probability that A & B both occur at the same time subtract P (A and B) in the above formula to remove any We _______ overlap.
Practice: Given a standard deck of cards, find each probability. 1. P(Drawing a red card) = 2. P(Drawing a king) = 3. P(Drawing a red card and king) = 4. P(Drawing a red card or a king) = P(Drawing a red card) + P(Drawing a king) – P(red and king) =
Practice: Given a standard deck of cards, find each probability. 5. P(Drawing a diamond) = 6. P(Drawing a face card) = 7. P(Drawing a diamond a face card) = 8. P(Drawing a diamond or a face card) = P(diamond) + P(face card) – P(diamond and face card) =
You have 5 minutes to complete the first several problems on the Mixed Practice. Whatever is not finished must be completed later.
Second Die Mixed Practice: Complete the area table for the outcomes of the event {roll two dice and find the sum} to find the probabilities. First Die 1 2 3 4 5 6 Mutually exclusive P(A or B) = P(A) + P(B) 1 2 3 4 5 6 7 1. P(sum is 2 or sum is 5) = 2 3 4 5 6 7 8 9 10 2. P(sum is a multiple of 2 and 5 6 7 8 9 10 11 sum is a multiple of 5) = 6 7 8 9 10 11 12 It’s an AND, so we’re good! Inclusive. P(A or B) = P(A) + P(B) – P(A and B) 3. P(sum is a multiple of 2 or sum is a multiple of 5) =
4. P(Both die are the same and sum < 8) = It’s an AND, so we’re good! Inclusive. Second Die First Die 1 2 3 4 5 6 7 P(A or B) = P(A) + P(B) – P(A and B) 5. P(Both die are the same or sum < 8) = 2 3 4 5 6 7 8 3 4 4 5 5 6 6 7 7 8 8 9 9 10 5 6 7 8 9 10 11 12
First Die 7. P(Odd Sum) = 8. P(sum = 7 and Odd Sum) = 9. P(sum = 7 | Odd Sum) = Second Die 6. P(sum = 7) = 1 2 3 4 5 6 7 8 3 4 4 5 5 6 6 7 7 8 8 9 9 10 5 6 7 8 9 10 11 12 Finish the rest of the Mixed Practice later.
Part 2: Complementary Events Recall the sample space for flipping a coin 3 times: S= HHH HHT HTH HTT TTH THT THH Let A = the event of getting at least 2 heads. Then P(A)= How would we describe all the elements of S other than A? At most one head (or at least 2 tails)
complement which means everything We call this AC (that is, A _____), but A. ____ Therefore, P(AC) = ______, so P(AC) + P(A) = _____ outcomes in the sample space Complement: The set of all _____ not included in the outcomes of event A. that are ____ 1 – P(A) Rule: P(AC) = ______ You have lovely teeth! Thanks! You have lovely hair! Okay, maybe we aren’t talking about these kinds of compliments.
Practice: Let’s return to the area model for the sum of 2 dice. Determine the following probabilities for the given events. 1) A = odd sum First Die AC = Not odd (or just even) P(AC) = 2) A = sum > 6 P(A) = AC = Second Die P(A) = 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 10 11 12 P(AC) =
Practice: Let’s return to the area model for the sum of 2 dice. Determine the following probabilities for the given events. First Die 1 2 3 4 5 6 7 8 9 4 5 6 7 8 9 10 11 12 P(A) = AC = Both dice are not the same Second Die 3) A = both dice are the same # P(AC) = 4) A = at least one die shows a 3 P(A) = AC = No threes P(AC) =
Finish the worksheets and PM 85, 88, 89, 91
The rest of the Mixed Practice
11. P(sum is a multiple of 3) = 12. P(sum is a multiple of 3 and sum is a multiple of 4) = First Die Second Die 10. P(sum is a multiple of 4) = 1 2 3 4 5 6 7 8 3 4 4 5 5 6 6 7 7 8 8 9 9 10 13. P(sum is a multiple of 4 | sum is a multiple of 3) = 14. P(sum is a multiple of 4 or sum is a multiple of 3) = 5 6 7 8 9 10 11 12
First Die 16. P(Sum > 7) = 17. P(Both die are the same and sum > 7) = Second Die 15. P(both die are the same) = 1 2 3 4 5 6 7 18. P(Both die are the same | sum > 7) = 19. P(Both die are the same or sum > 7) = 2 3 4 5 6 7 8 3 4 4 5 5 6 6 7 7 8 8 9 9 10 5 6 7 8 9 10 11 12
First Die Second Die 20. P(At least one die is a 4) = 21. P(Sum = 9) = 1 2 3 4 5 6 7 22. P(At least 1 die is a 4 and sum = 9) = 23. P(Sum = 9 | at least 1 die is a 4) = 2 3 4 5 6 7 8 3 4 4 5 5 6 6 7 7 8 8 9 9 10 5 6 7 8 9 10 11 12
Pencil, red pen, highlighter, GP notebook, textbook, calculator Solve for all solutions of x. 1) 2) +2 factor +1 ZPP +4 + 4 +1 +2 +1 +1 +1 total: +2 +1 +1 +2
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