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Elementary Statistics and Inference 22S:025 or 7P:025 Lecture 18 1 Elementary Statistics and Inference 22S:025 or 7P:025 Chapter 14 2 14. More About Chance A. Listing All Possibilities Suppose two dice are thrown, the possible ways the dice could fall is 36. 3 1 14. More About Chance (cont.) 4 14. More About Chance (cont.) Using this chart we can list the probabilities (chances) of obtaining a total on the dice from 2 to 12. X (Total) P(X) 2 1/36 3 2/36 4 3/36 5 4/36 6 5/36 7 6/36 8 5/36 9 4/36 10 3/36 11 2/36 12 1/36 P( X ) = Total ways of success Total ways Event can occur Sum=1.00 5 14. More About Chance (cont.) Graphically 6/36 5/36 4/36 3/36 2/36 1/36 0 1 2 3 4 5 6 7 8 9 10 11 12 If we had 3 dice, the number of possible ways the dice could fall is 6 x 6 x 6 = 216. 6 2 14. More About Chance (cont.) B. The Addition Rule Two events (outcomes) are mutually exclusive when the occurrence of one prevents the occurrence of the second event – one excludes the other other. Example: If a deck is used to select a card, the outcome can be a “heart” or a “spade” – they are mutually exclusive. 7 14. More About Chance (cont.) If events are mutually exclusive, to find the probability (chance) or either happening you add the probabilities. P ( heart) + P (spade) = 13 13 26 1 + = = 52 52 52 2 Example: Someone throws a pair of dice. Is the chance 1 1 1 of at least one Ace + = ? See Figure 1 6 6 3 ⎛ 1 1 1 11 ⎞ - ⎜ + − = ⎟. ⎝6 6 36 36 ⎠ If two events are not mutually exclusive, do not add the chances – the sum is too big. 8 14. More About Chance (cont.) Exercise Set B (pp. 242-243) #2, 3, 4, 5 #5. (p.243) – A box contains 10 tickets numbered 1-10. Five draws will be made at random with replacement from this box box. True or False: there are 5 chances in 10 of getting at least one “7” in the five draws. False – the draws are not mutually exclusive P (7) = 1 for each draw. 10 9 3 14. More About Chance (cont.) #6. A number is drawn at random from a box. There is a 20% chance for it to be 10 less, and a 10% chance for it to be 50 or more. The chance of the number to be between 10 and 50 – 70%. 20% 70% 10% 10 50 10 14. More About Chance (cont.) C. Two Frequently Asked Questions Question: 1) What is difference between mutually exclusive and independent? 2) When do I add probabilities, and when do I multiply probabilities? 11 14. More About Chance (cont.) Answer: Two events are mutually exclusive if the occurrence of one prevents the other from happening. Two events are independent if the occurrence of one does not change the chances (probability) of the other. The addition rule finds the chance (probability) that at least one of two things happens -- add probabilities if events are mutually exclusive If need to determine probability that both independent events happen – multiple the probabilities. (for dependent events, the multiplication rule uses conditional probabilities.) 12 4 14. More About Chance (cont.) Example 6: p. 244 A die is rolled 6 times; a deck of cards is shuffled. a) The chance the first roll is an ace or the 11 1 1 1 last roll is a six is 36 = 6 + 6 − 36 b) The chance that the first roll is an ace and the 1 1 1 = × last roll is an ace is 36 6 6 13 14. More About Chance (cont.) c) The chance that the top card is the ace of spades or the bottom card is the ace of 1 1 1 spades is 52 + 52 = 26 d) The chance the top card is an ace of spades and the bottom card is an ace of spades is 0 14 14. More About Chance (cont.) p. 247 – Exercise 3 3. A deck of cards is shuffled. True or false, and explain briefly: a) b) c) d) e) f) The chance that the top card is the jack of clubs equals 1/52. The chance that the bottom card is the jack of diamonds equals q 1/52. The chance that the top card is the jack of clubs or the bottom card is the jack of diamonds equals 2/52. The chance that the top card is the jack of clubs or the bottom card is the jack of clubs equals 2/52. The chance that the top card is the jack of clubs and the bottom card is the jack of diamonds equals 1/52 x 1/52. The chance that the top card is the jack of clubs and the bottom card is the jack of clubs equals 1/52 x 1/52. 15 5 14. More About Chance (cont.) p.247 – Exercise 4 4. The unconditional probability of event A is ½. The unconditional probability of event B is 1/3. Say whether each of the following is true or false, and explain briefly. a) b) c) d) e) f) The chance that A and B both happen must be ½ x 1/3 = 1/6. If A and B are independent, p the chance that they y both happen must be ½ x 1/3 = 1/6. If A and B are mutually exclusive, the chance that they both happen must be ½ x 1/3 = 1/6. The chance that at least one of A or B happens must be ½ + 1/3 = 5/6. If A and B are independent, the chance that at least one of them happens must be ½ + 1/3 = 5/6. If A and B are mutually exclusive, the chance that at least one of them happens must be ½ + 1/3 = 5/6. 16 14. More About Chance (cont.) Exercise Set D – p. 250 17 14. More About Chance (cont.) Exercise Set D – p. 250 4. a) A die is rolled 3 times. What is the chance of getting at least one ace? b) Same, with 6 rolls? c) Same, with 12 rolls? 5. A pair of dice is rolled 36 times. What is the chance of obtaining at least one double-ace? 18 6 14. More About Chance (cont.) Review Exercises – pp. 252-253 #1 – 10 When a die is rolled, each of the 6 faces is equally likely to come up. A deck of cards has 4 suits (clubs, diamonds, hearts, spades) with 13 cards in each suit – 2 3 2, 3, …, 10 10, jack jack, queen queen, king king, ace ace. See pp pp. 222 and 226. 1. A pair of dice are thrown. a) Find the chance that both dice show 3 spots. b) Find the chance that both dice show the same number of spots. 19 20 14. More About Chance (cont.) 2. In the game of Monopoly, a player rolls two dice, counts the total number of spots, and moves that many squares. Find the chance that the player moves 11 squares (no more and no less. 3. True or false, and explain: a) If a die is rolled three times, the chance of getting at least one ace is 1/6 + 1/6 + 1/6 = ½. b) If a coin is tossed twice, the chance of getting at least one head is 100%. 21 7 22 14. More About Chance (cont.) 4. Two cards will be dealt off the top of a well-shuffled deck. You have a choice: i. ii ii. To win $1 if at least one of the two cards is a queen. T win To i $1 if th the fifirstt iis a queen. Which option is better? Or are they equivalent? Explain. 23 14. More About Chance (cont.) 5. The chance of A is 1/3; the chance of B is 1/10. True or false, and explain: a) If A and B are independent, they must also be mutually exclusive exclusive. b) If A and B are mutually exclusive, they cannot be independent. 24 8 14. More About Chance (cont.) 6. One event has chance ½, another has chance 1/3. Fill in the blanks, using one phrase from each pair below, to make up two true sentences. Write out both sentences. “If If you want to find the chance that (i) will happen, check to see if they are (ii) . If so, you can (iii) the chances.” i. ii. iii. at least one of the two events, both events independent, mutually exclusive add, multiply 25 14. More About Chance (cont.) 7. Four draws are going to be made at random with replacement from the box 1 2 2 3 3 Find the chance that “2” is drawn at least one time. 8. Repeat exercise 7, if the draws are made at random without replacement. 26 14. More About Chance (cont.) 9. One ticket will be drawn at random from each of the two boxes shown below: (A) a) b) c) 1 2 3 (B) 1 2 3 4 Find the chance that: The number drawn from A is larger than the one from B. The number drawn from A equals the one from B. The number drawn from A is smaller than the one from B. 27 9 14. More About Chance (cont.) 10. There are two options: i. ii. A die will be rolled 60 times. Each time it shows an ace or a six, you win $1; on the other rolls, you win nothing. Sixty draws will be made at random with replacement from the box 1 1 1 0 0 0 On each draw, you will be paid the amount shown on the ticket, in dollars. Which option is better? Or are they the same? Explain briefly. 28 10