Quiz Probability 1.) A telemarketer knows from past experience that when she makes a call, the probability that someone will answer the phone is 0.20. What is probability that the next two phone calls she makes will both result in someone answering the phone? (A) 0.04 ( 0.20 (C) 0.24 (D) 0.40 (E) 0.80 2.) If P( A ) = 0.34 and P( AÈ = 0.71, which of the following is false? (A) PB ( ) = 0.37, if A and B are mutually exclusive. ( PB ( ) = 0.561, if A and B are independent. (C) P( can t be determined if A and B are neither mutually exclusive nor independent. (D) P( AÇ = 0.191, if A and B are independent. (E) P( A B ) = 0.34, if A and B are mutually exclusive. 3.) Experience has shown that a medical test for a certain disease will show a false positive 5% of the time and will correctly show a positive result 80% of the time when the individual being tested truly does have the disease. Suppose that a random sample of 3 individuals is given the test and that all three of the individuals have the disease that is being tested for. What is the probability that at least one of the tests will show a negative result? (A) 0.032 ( 0.128 (C) 0.488 (D) 0.512 (E) 1.650 4.) If X and Y are two independent events, P( X ) = 0.30, PY ( ) = 0.50. Find P( XÈ Y). (A) 0.15 ( 0.30 (C) 0.50 (D) 0.65 (E) 0.80 74
5.) A die is loaded so that the number 1 comes up twice as often as any other number. What is the probability of rolling an odd number? (A) 1 4 ( 1 2 (C) 4 7 (D) 3 5 (E) 2 3 c c 6.) Suppose that A and B are two independent events with P( A ) = 0.7 and PB ( ) = 0.4. Find P( AÇ. (A) 0.18 ( 0.28 (C) 0.55 (D) 0.70 (E) 0.90 7.) Given: P( A ) = 0.3 and PB ( ) = 0.6, P( AÇ = 0.2. Find the probability of B given A. (A) 1 3 ( 2 3 (C) 2 5 (D) 3 5 (E) 49 50 75
8.) Given two events Land F, if PL ( ) = 0.58, PF ( ) = 0.50 and PL ( Ç F) = 0.31, what is PL ( ÈF)? Use both formula and the probability table to answer the question. 9.) Suppose that PE ( ) = 0.7, PF ( ) = 0.6 and PE ( Ç F) = 0.54. Calculate PE ( F) and PF ( E ). 10.) A certain university has 10 vehicles available for use by the faculty and staff. Six of these are vans and four are cars. On a particular day, only two requests for vehicles have been made. Suppose that two vehicles to be assigned are chosen at random from among the 10 vehicles. a.) Let E denote the event that the first vehicle assigned is a van. What is the value of P( E )? b.) Let F denote the event that the second vehicle assigned is a van. What is the value of PF ( E )? c.) Use the results of (a) and (b) to calculate PF ( Ç E)? 76
11.) A spinner contains five sections of equal area each with a different color (red, yellow, blue, green, and purple.) If the spinner is spun 5 times, what is the probability that: a.) all 5 spins will result in either red or yellow? b.) none of the spins will result in blue? c.) at least one spin will result in purple? d.) four spins will result in green and one spin will result in yellow? 12.) McDonalds advertises that there is one winner in every 4 game pieces in its version of the Monopoly Game. Suppose that you visit McDonalds three times, collecting a single game piece each time. What is the probability that: a.) you win at least once? b.) you win at least twice? c.) you win all three times, given that you win at least twice? 13.) [APSTATSMC2014-11] A carnival game allows the player a choice of simultaneously rolling two, four, six, eight, or ten fair dice. Each die has six faces numbered 1 through 6, respectively. After the player rolls the dice, the numbers that appear on the faces that land up are recorded. The player wins if the greatest number recorded is 1 or 2. How many dice should the player choose to roll to maximize the chance of winning? (A) Two ( Four (C) Six (D) Eight (E) Ten 77
14.) [The Tenafly Coin Problem] A game is played by tossing three coins, and the outcome is hidden before you make a bet. The information whether there are at least two heads up in the outcome is given before your decision. You win $1 when there are three heads up. You decide to pay $20 to play the game a 100 times. For each game, you make a bet only when the outcome of the game contains at least two heads up. How much would you expect to make at the end of 100 games if a.) the coins are fair coins. b.) each coin has a 60% probability of being heads up for each tossing. 78
Answers: 1.) A. P ( two Calls ) = 0.2(0.2) = 0. 04, assumed that the two calls are independent. 2.) E, P( A = P( A) implies independent. For choice A, if A and B are mutually exclusive, P ( AÈ = P( A) + P( P( = 0.37, and P ( A) P( = 0.34(0.37) = 0.1258 ¹ 0. For choice B, if A and B are independent, P ( AÈ = P( A) + P( -P( A) P( P( = 0.56. 3.) C. Let P = Result is Positive and D = The person has the disease, then c P( P D ) = 0.05, P( P D) = 0.8. P( three positive D) = (0.8) 3 = 0. 512 and P ( atleast onenegative D) = 1- (0.8) 3 = 0.488 4.) D, P( XÈ Y) = P( X ) + PY ( )-P( XÇ Y) = 0.3+ 0.5-0.3 0.5 = 0.8-0.15= 0.65 5.) C. Let p be the probability of other numbers except 1, then 1 1= 2p+ p+ p+ p+ p+ p= 7p, so p=. Podd ( ) = 2p+ p+ p= 4p= 4 7 7 c c P( AÇ = P( APB ) ( ) = 1 -P( A ) 1 - PB ( ) = (1-0.7)(1-0.4) = 0.18 6.) A, ( )( ) 7.) B, P( AÇ 0.2 2 P( B A) = = =. P( A) 0.3 3 8.) PL ( È F) = PL ( ) + PF ( )-PL ( Ç F) = 0.58+ 0.50-0.31= 0.77. From the given, the probability table can be constructed (the boxed numbers are given): L \ F F c F Total L 0.31 0.58-0.31= 0.27 0.58 c L 0.50-0.31= 0.19 0.42-0.19= 0.23 1-0.58= 0.42 Total 0.50 0.27+ 0.23= 0.5 1 From the table PL ( È F) = 1-0.23= 0.77. 9.) 0.9, 0.771 79
10.) 6 3 P( E ) = =, 10 5 5 P( F E ) =, 9 3 5 1 P( FÇ E) = P( F E) P( E) = = 5 9 3 11.) a.) b.) c.) d.) 12.) a.) b.) c.) 2 Predoryellow ( ) = æ ö ç è 5ø æ 4ö Pnotblue ( ) = ç è 5ø æ 4ö Patleastonepurple ( ) = 1- ç è 5ø 5 1 1 P(4greenN 1 yellow) C æ ö æ ö è 5ø è 5ø 5 = 5 4ç ç æ 3ö 37 Patleastonce ( ) = 1- ç = è 4ø 64 3 5 4 2 3 Patleasttwice 1 3 1 5 ( ) = C æ 3 2ç ö æ ç ö + æ ç ö = è 4ø è 4ø è 4ø 32 æ 1ö ç 4 1 P(3 times atleasttwice) = è ø = 2 3 æ 1ö æ 3ö æ 1ö 10 3C2ç ç + ç è 4ø è 4ø è 4ø 3 13.) A. Two out of six faces are the favorable outcomes. So, 2 10 1 æ 1ö æ 1ö P(1 D) =, P(2 D) = ç,, P(10 D) = ç 3 è 3ø è 3ø 80
14.) a.) This is an equally likely problem. Four outcomes have two heads and one outcome 1 has three heads. So, P ( 3H 2H) =, and the expected winning amount of playing 100 4 æ 1ö times is $ 100ç - $20= $25- $20= $ 5. è 4ø b.) 81