Tiêu đề MSTE 5 Solutions.pdf ✅

05 Algebra: Probability and Statistics Solutions ▣ 1. The lotto uses numbers 1 to 42. A winning ticket uses 6 different numbers in any order. What is the probability of winning the lotto with only one ticket? [SOLUTION] Since the arrangement is not considered, then 1 42C6 = 1 5245786 or 1.9063 × 10−7 ▣ 2. If you roll a pair of dice, what is the probability of getting a sum of 9? [SOLUTION] The possible outcomes that show a sum of 9 are (3,6), (4,5), (5,4), (6,3). There are 6 × 6 = 36 total possible ways in rolling two dice. 4 36 = 1 9 ▣ 3. A coin is tossed twice. What is the probability that at least one head occurs? [SOLUTION] Using the probability of the complement, P(at least one head) = 1 − P(no heads) = 1 − ( 1 2 ) 2 = 3 4 ▣ 4. The probability that a doctor correctly diagnoses a particular illness is 0.7. Given that the doctor makes an incorrect diagnosis, the probability that the patient enters a lawsuit is 0.9. What is the probability that the doctor makes an incorrect diagnosis and the patient sues? [SOLUTION] Let A be the event of a correct diagnosis and B for suing. P(A) = 0.7 P(B|A c ) = 0.9 P(A c ∩ B) = P(B ∩ A c ) = P(A c ) ⋅ P(B|A c ) P(A c ∩ B) = (1 − 0.7)(0.9) = 0.27
▣ 5. There are 5 green and 7 red balls. Two balls are selected one by one without replacement. Find the probability that the first is green and the second is red. [SOLUTION] 5 12 × 7 11 = 35 132 ▣ 6. 100 raffle tickets are sold at Php 5.00. If there is only one winner of Php 100, find the expected value of purchasing one ticket. [SOLUTION] (−5) + 100 × 1 100 = Php 4 loss ▣ 7. In a certain town, the probability that a person has blue eyes is 0.25. The probability that a person has blonde hair is 0.4; and the probability that a person has both blue eyes and blonde hair is 0.14. If a certain person has blue eyes, what is the probability that the person also has blonde hair? [SOLUTION] Blue Eyes Not Blue Eyes Total Blonde Hair 0.14 0.4 − 0.14 = 0.26 0.4 Not Blonde Hair 0.25 − 0.14 = 0.11 0.6 − 0.11 = 0.49 1 − 0.4 = 0.6 Total 0.25 1 − 0.25 = 0.75 1 0.14 0.25 = 0.56 ▣ 8. In the previous question, if a person has blonde hair, what is the probability that the person also has blue eyes? [SOLUTION] 0.14 0.4 = 0.35 ▣ 9. A family has two children. If one of them is a boy, what is the probability that the other is also a boy? [SOLUTION] P(the other is also a boy|one is a boy) = P(two boys) P(at least one boy)
= ( 1 2 ) 2 1 − (1 − 1 2 ) 2 = 1 3 Verify: Possible arrangement of children = BB, BG, GB, GG. There are 3 cases where only 1 has both boys. Therefore, the probability is 1/3.