Math 511 Exam #1. Show All Work No Calculators

Transcription

1 Math 511 Exam #1 Show All Work No Calculators 1. Suppose that A and B are events in a sample space S and that P(A) = 0.4 and P(B) = 0.6 and P(A B) = 0.3. Suppose too that B, C, and D are mutually independent events and P(C) = 0.7 and P(D) = 0.8 What is the value of: [Express all answers as decimal values.] (a). P(A B) = Answer: 0.18 P( A B) = P( A B)P(B) = = 0.18 (b). P(A B) = Answer: 0.55 ( ) ( ) P(A B) = P A B P B ( ) P( A B) = P A 1 P(B) = = 0.55 (c). P(A B ) = Answer: 0.82 P(A B ) = 1 P( A B) = = 0.82 (d). P( C D) = Answer: 0.94 P( C D) = P(C) + P(D) P( C D) = = 0.94 (e). What is the probability that at least one of B, C, or D occurs? Answer: We want 1 P B C D ( ) = 1 P B ( )P C ( )P D ( ) = = 0.976

2 2. Assuming the validity of Benford s law, (a). The probability that a randomly chosen power of 2 begins with one of the digits 6 or 7 is the same as the probability it begins with the digit d. Determine the value of d and justify your answer. P(d) = P(6) + P(7) = log log 8 7 = log = log 8 6 = log 4 3. But P(3) = log 4 3, and so d = 3. (b). What is the probability that a randomly chosen power of two begins with a 5 and its second digit is not a 3? [Express your answer as the log of a single number.] Let B5 denote that the number begins with a 5 and S3 that the second digit is 3. The answer is log We want P B5 S3 = P B5 ( ) P( B5 S3) = P( first digit is 5 ) P ( first two digits are 53 ) = log 6 5 log = log = log In your pocket you have 6 fair coins and 4 biased coins. When flipped, the biased coins show heads 80% of the time. You choose a coin and flip it. Given that the coin shows heads, what is the probability that it is the fair coin? [Show all work and simplify your answer to a fraction in lowest terms.] Let F denote that the coin is fair, B that the coin is biased. P(F H ) = P( H F)P(F). P(H ) However, P( H ) = P( H F)P( F) + P( H B)P( B) = = So, P(F H ) = P( H F)P(F) = = 30 P(H ) =

3 4. (a). Suppose that a box contains 8 each of red, blue, green, yellow and orange marbles. If 10 marbles are drawn at random from the box (without replacement), then what is the probability that they consist of 4 each of two different colors and two of a third color? (Example: 4 red, 4 blue and 2 green OR 4 green 4 blue and 2 red) (b). What is the probability that a randomly chosen subset of {a, b, c, d, e, 1, 2, 3, 4, 5, 6} will contain exactly 3 digits, but any number of letters? Examples: {2, 4, 5}, {1, 5, 6, a, b}, {2, 3, 4, c}. [Express as a fraction in lowest terms.] = = = distinct marbles are placed randomly into 9 distinct boxes numbered 1 through 9. What is the probability that the first box contains exactly 4 marbles, the second box has exactly 5 marbles, and the third box is empty? Simply choose some 4 of the original 16 for the first box, then some 5 of the remaining 12 for the second box and then there are 7 marbles left to place into 6 boxes. 6. A drawer contains 63 socks, 7 each of 9 different colors. Six socks are chosen at random from the drawer. What is the probability that at least two of the socks have the same color?

4 7. (a). Suppose that an archer shoots 20 arrows at a target. The results of the shots are mutually independent and the archer hits the target 80% of the time. Let X denote the number of times the archer hits the target. (i). P( X = 7) = (no need to simplify binomial coefficients or powers.) ( ) = 20 7 P X = 7 ( 0.8)7 0.2 ( ) 13 (ii). E(X) = 16 (exact value as a decimal.) E(X) = = 16 (b). According to the Binomial Theorem, the expansion of (x + y) n is given by (x + y) n = n k =0 n k xk y n k 8. A 1000-sided die has 4 red faces and 996 green faces. Suppose that the die is rolled times. Then (a). Determine an expression for the exact probability that a red face appears on at least one roll. This is 1 minus the probability that no red faces occur. So we get, ( ) = (b). What is the approximate value of the answer in (a) in terms of the constant e? [Explain your answer.] ( ) = = ( e 4 ) 12 = 1 e 2 Or, equivalently, ( ) = = e 2 Note: This evaluates to about ; the exact value is

5 9. Suppose that a carton of 12 eggs contains 5 that are rotten. If the eggs are tested one at a time, what is the probability that the 3 rd rotten egg is the 6 th egg tested? Let A be the event that of the first five eggs tested, 2 were rotten and 3 were good. Let B be the event that the 6 th egg tested is rotten. Note that P( A) = and P( B A) = 3 7. Then we want the value of P( A B) = P( B A)P(A) =