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n! n Cr n! Quiz 10-1 thru 10-4 r!(n r )!nP (n r )! r 1. Is counting the number of ways to form a 3 person committee out of 10 people a combination or a permutation? 2. How many different licence plates are possible if there are 2 letters, followed by 4 numbers, then 2 letters? 3. What is the probability of dealing a king out of a well-shuffled deck of cards? 4. If P(A) = 0.5, P(B) = 0.4, and P(A or B) = 0.8. Are the two events A and B, disjoint or overlapping? What is P(A and B)? 5. How many ways can you arrange 10 candidates into the positions of President, Vice President, and Secretary ? 10-5 Conditional and Unconditional Probability. (Sequential events) The Probability of Sequential Events The probability of event A happening followed by event B. Event A: rolling a 5 on a single die. Event B: rolling a 6 on a single die. Event A: drawing a red marble out of a bag of marbles (containing more than one red marble and some other colors) Event B: drawing another red marble out of the same bag of marbles (without replacement) How are these two situations different? Vocabulary Conditional Probability: the probability of an event depends upon the outcome of a previous event occurring (drawing a red ball on the second draw out of a bag of colored marbles). Unconditional Probability : the probability of an event does not depend upon the outcome of a previous event occurring (the probability of drawing a king on the second card dealt when the previously dealt card was shuffled back into the deck before drawing the second card (with replacement). Conditional vs. Unconditional Probability. 150 raffle tickets were sold for a mall gift certificate. 200 tickets were sold for a movie pass. What is the probability that you win both? Does the probability of winning the mall gift certificate depend upon whether or not the movie pass was won or not? Winning the gift certificate or winning the movie pass are independent events. Dependent vs. Independent Events A BMX bicycle race as 8 starting positions called “slots”. What is the probability of drawing slot #8 for 3 races is a row? Does the probability of drawing slot #8 on the second race depend upon the slot that was drawn in the previous race? Drawing a specific slot for subsequent races are independent events. Dependent vs. Independent Events Which is it, dependent or independent? For fund raiser, 100 tickets are sold for dinner for two at Chile’s Restaurant, and 200 tickets are sold for a booklet of movie passes. You buy 3 of the dinner and 5 of the movie raffle tickets. What is the probability of winning both? What is the probability of rolling a pair of dice and having their sum be 7 ? Your turn: which is it, dependent or independent? 1. For fund raiser, 100 tickets are sold for dinner for two at at Chile’s Restaurant, and 200 tickets are sold for a booklet of movie passes. You buy 3 of the dinner and 5 of the movie raffle tickets. What is the probability of winning the movie booklet? 2. A red marble is drawn from a bag of marbles. What is the probability of drawing another red marble (without replacement) ? 3. A card is dealt face up from a deck of cards. What is the probability that the second card dealt will be a king (without replacing the first card)? 4. What is the probability of rolling a 5 then rolling another 5 using the same die? Probability of Independent Events P(A and B) where events “A” and “B” are independent: P(A and B) = P(A) * P(B) What is the probability of rolling a 5 then rolling another 5 using the same die? 1 1 1 P (“5” and “5”) = ? * 0.027 6 6 36 Independent Probability. For fund raiser, 100 tickets are sold for dinner for two at at Chile’s Restaurant, and 200 tickets are sold for a booklet of movie passes. You buy 3 of the dinner and 5 of the movie raffle tickets. What is the probability of winning both? 3 5 15 P (“dinner” and “movie”) = ? * 100 200 20000 3 0.00075 4000 Independent Probability. Your turn: 5. What is the probability of rolling “snake eye’s” (two 1’s) followed by another “snake eye’s” on the second roll of a pair of “fair” dice ? 6. Events “A” and “B” are independent events. P(A) = 0.9 P(B) = 0.8 P(A and B) = ? Geometric Probability: Assumming that at an arrow randomly hits anywhere in the four square area, what is the probability of hitting in the #1 square with the first dart and the #4 square with the second dart? What would be the 1 2 probability of hitting in order squares #1 through #4 with 4 darts thrown one after the other? 3 4 Probability 7. What is the probability of hitting the outer ring on with the first dart and the “bull’s eye” with the second dart? 8. What is the probability hitting the “bull’s eye” on the first try then not hitting the “bull’s eye” on the second try? 3 2 Dependent Probability. A card is dealt face up from a deck of cards. What is the probability that the second card dealt will be a king? The probality of getting a king on the second card depends upon whether there was a king drawn on the first card. Case 1: king was drawn Case 2: king not drawn first (without first (no replacement replacement) of the card) 3 4 P(2 king) nd P(2 nd king) 51 51 The probability of the second event depends upon whether there was a king drawn on the first card. Probability of Dependent Events P(A and B) where events “A” and “B” are dependent: P(A and B) = P(A) * P(‘B’ given that event ‘A’ occurred) P(A and B) P( A) * P B A Event B: rolling a 6 on a single die. A card is dealt face up from a deck of cards. What is the probability that the first card is not a king and the second card dealt is a king? 48 4 P(1 not a king) st P(2nd is a king given that 1st as not a king) 52 51 P(A and B) P( A) * P B A 48 4 * 52 51 192 0.072 2652 Probability of Dependent Events P(A and B) = P(A) * P(‘B’ given that event ‘A’ occurred) A bag contains 3 red, 3 blue, and 3 green marbles. You and two friends pick a marble from the bag (without replacing). What is the probability that you will each have a different color of marble? # of ways to choose a marble 9 First draw: P(event ) total # of ways that 9 marbles can be chosen 9 Second P(event) # of ways to choose one marble out of 2 colors 6 draw: total # of ways that 8 marbles can be chosen 8 # of ways to choose one marble out of 1 color 3 Third P(event) total # of ways that 7 marbles can be chosen 7 draw: 9 6 3 18 P(event) * * 0.321 9 8 7 56 Independent Probability. A bag of marbles contains 3 red, 4 blue, and 6 green marbles. 3 What is the probability of drawing a red marble 1st? 13 10 What is the probability of not drawing a red marble 1st? 13 What is the probability of drawing a red marble 1st, then not drawing a red marble 2nd (with replacement)? Can I use this equation? P(A and B) P( A) * P B A 3 10 30 * 0.18 13 13 169 Your turn: 9. What is the probability of drawing an ace, then (without replacing the ace) drawing a king out of well-shuffled deck of cards? 10. Events “A” and “B” are dependent events. P(A) = 0.5 P(B given that A occured) = 0.4 P(A and B/A) = ? 11. The probability of dealing a heart on the first card is: P(heart) = 0.25 What is the probability of drawing a heart 1st followed by not drawing a heart on the 2nd card (no replacement)? Total possible outcomes of rolling 2 dice: P( rolling a sum of 5 using 2 dice) = ? die roll 1 2 3 4 5 6 1 1+1=2 2+1=3 3+1=4 4+1=5 5+1=6 6+1=7 2 1+2=3 2+2=4 3+2=5 4+2=6 5+2=7 6+2=8 3 1+3=4 2+3=5 3+3=6 4+3=7 5+3=8 6+3=9 4 1+4=5 2+4=6 3+4=7 4+4=8 5+4=9 6+4=10 5 1+5=6 2+5=7 3+5=8 4+5=9 5+5=10 6+5=11 6 1+6=7 2+6=8 3+6=9 4+6=10 5+6=11 6+6=12 4/36 = 1/9 = 0.1111 Your turn: 12. What is the probability of having the sum of two fair dice being a 7? 13. What is the probability of being dealt a queen then an ace if the queen is put back into the deck and shuffled before you are dealt the second card (with replacement)? 14. What is the probability of being dealt a queen then an ace if the queen is not put back into the deck before you are dealt the second card (without replacement)? Probability In the movie “Tron” a game is played where two contestants are on circular stands. One player throws a ball toward the player’s circle. If that player doesn’t catch the ball, it will hit one of the rings he is standing on and the ring will disappear. He needs to either catch the ball or get off the ring before it disappears. If a player is standing on a ring when it disappears he will fall to his death. 125 15. What is the probability of a 75 player still being alive if he doesn’t catch the ball, doesn’t 25 move and he is standing on the blue ring? Dependent/independent vs. Disjoint/overlapping events. Dependent/Independent events deal with sequential events. Drawing a king then a 5. P(A and B) P( A) * P B A Drawing a red then a blue marble Disjoint/overlapping events deal with an event that can be categorized in more than one way. Being a girl with brown hair in a group of boys and girls of different hair colors. P(A or B)= P(A) + P(B) – P(overlap) Summary Disjoint/overlapping events deal with an event that can be categorized in more than one way. P(boy or girl) ? P(A or B)= P(A) + P(B) – P(overlap) (Blonde (Girls) 2 3 0 5 (boys)Hair) Amber Maria Angelica 5 5 5 5 Bill 2 2 1 3 P(boy or blond hair) ? Frank 5 5 5 5

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