Module 11 Written Assignment - Solutions 1 1. A fair die is rolled five times. a. What is the probability of getting five 6's in a row? b. What is the probability of getting all even numbers? 2. Network Reliability: In the network shown below, the links between computers can be either up or down. The reliability of each link is as shown in the diagram. The status of each link is independent of the status of the other links. What is the probability that there is a functioning set of links connecting computer B to computer C? A B Uptime = 85% Uptime = 93% Uptime = 96% C 3. Consider drawing one card from a standard 52-card deck. Define four events as follows: A={Hearts } , B={ cards } , C={ Aces } , and D={Queen of Hearts ,Queen of Spades } . Prove which pairs of these events are independent, and which are dependent. 4. An urn contains 7 red balls and 5 green balls. Two balls are drawn at random from the urn. What is the probability of getting two balls of the same color if: (a) the first ball is replaced before the second ball is drawn. (b) the first ball is not replaced before the second ball is drawn. Module 11 Written Assignment - Solutions 2 5. Communication Channel Reliability: Consider a communication channel over which a 0 or a 1 must be transmitted. Suppose that the probability that the bit to be sent is a 0 is 0.5, and the probability that a 1 is to be sent is also 0.5. Also suppose that due to noise the probability that a 0 is changed to a 1 during transmission is 0.06 and the probability that a 1 is changed to a 0 is 0.03. Suppose that a 0 is received. What is the probability that a 0 was sent? Module 11 Written Assignment - Solutions 1 1. A fair die is rolled five times. a. What is the probability of getting five 6's in a row? b. What is the probability of getting all even numbers? a) If the dice is rolled once the probability will be chance it lands on 6. In 5 rolls be ( ) b) We have 3 even numbers. Therefore the probability will be . In 5 rolls it become ( ) 2. Network Reliability: In the network shown below, the links between computers can be either up or down. The reliability of each link is as shown in the diagram. The status of each link is independent of the status of the other links. What is the probability that there is a functioning set of links connecting computer B to computer C? B Uptime = 85% A Uptime = 93% Uptime = 96% C P = (0.5*0.93) + (0.5*0.85*0.96) = 0.873 3. Consider drawing one card from a standard 52-card deck. Define four events as follows: * +, * +, * +, and * +. Prove which pairs of these events are independent, and which are dependent. (A, C), (A, D), (B, C) and (B, D) are dependent and all others are independent 4. An urn contains 7 red balls and 5 green balls. Two balls are drawn at random from the urn. What is the probability of getting two balls of the same color if: (a) the first ball is replaced before the second ball is drawn. (b) the first ball is not replaced before the second ball is drawn. Module 11 Written Assignment - Solutions 2 a) Number of balls = 12 1st turn get red ball; probability = 7/12 2nd turn = 7/12 Probability of getting red in 2 turns = For green balls = Therefore probability with replacement = 0.340 + 0.173 = 0.513 b) Probability of getting red balls = Probability of getting blue balls = Probability without replacement = 5. Communication Channel Reliability: Consider a communication channel over which a 0 or a 1 must be transmitted. Suppose that the probability that the bit to be sent is a 0 is 0.5, and the probability that a 1 is to be sent is also 0.5. Also suppose that due to noise the probability that a 0 is changed to a 1 during transmission is 0.06 and the probability that a 1 is changed to a 0 is 0.03. Suppose that a 0 is received. What is the probability that a 0 was sent? A - event that 0 is received B - event that 0 is sent P(BIA) = ( ) ( ) P(A) = (0.5*0.03) + (0.5*0.94) = 0.485 ( ( ) ) Module 11 Written Assignment - Solutions 1. A fair die is rolled five times. a. What is the probability of getting five 6's in a row? b. What is the probability of getting all even numbers? a) If the dice is rolled once the probability will be chance it lands on 6. In 5 rolls be ( ) b) We have 3 even numbers. Therefore the probability will be . In 5 rolls it become ( ) 2. Network Reliability: In the network shown below, the links between computers can be either up or down. The reliability of each link is as shown in the diagram. The status of each link is independent of the status of the other links. What is the probability that there is a functioning set of links connecting computer B to computer C? A B Uptime = 85% Uptime = 93% Uptime = 96% C P= (0.5*0.93) + (0.5*0.85*0.96) = 0.873 3. Consider drawing one card from a standard 52-card deck. Define four events as follows: A={Hearts } , B={ cards } , C={ Aces } , and D={Queenof Hearts ,Queen of Spades } . Prove which pairs of these events are independent, and which are dependent. A - number of outcomes of getting hearts B - number of outcomes of getting black cards C - number of outcomes of getting aces D - number of outcomes of getting queen of hearts and queen of spades P(A)=13/52; P(B) = 26/52=1/2; P(B) = 4/52 = 1/13; P(D) = 2/52 = 1/26 Two vents are independent if 1 Module 11 Written Assignment - Solutions P ( A B ) =P ( A )P ( B ) ; P ( A B ) =0 ; P ( A C )= 2 1 1 1 1 ; P ( A D )= ; P ( B C )= ; P ( B D ) = ; P ( C 52 52 26 52 Therefore, (A, C), (A, D), (B, C), (B, D)are dependent and all others independent 4. An urn contains 7 red balls and 5 green balls. Two balls are drawn at random from the urn. What is the probability of getting two balls of the same color if: (a) the first ball is replaced before the second ball is drawn. (b) the first ball is not replaced before the second ball is drawn. 5. Communication Channel Reliability: Consider a communication channel over which a 0 or a 1 must be transmitted. Suppose that the probability that the bit to be sent is a 0 is 0.5, and the probability that a 1 is to be sent is also 0.5. Also suppose that due to noise the probability that a 0 is changed to a 1 during transmission is 0.06 and the probability that a 1 is changed to a 0 is 0.03. Suppose that a 0 is received. What is the probability that a 0 was sent? Module 11 Written Assignment - Solutions 1 1. A fair die is rolled five times. a. What is the probability of getting five 6's in a row? b. What is the probability of getting all even numbers? a) If the dice is rolled once the probability will be chance it lands on 6. In 5 rolls be ( ) b) We have 3 even numbers. Therefore the probability will be . In 5 rolls it become ( ) 2. Network Reliability: In the network shown below, the links between computers can be either up or down. The reliability of each link is as shown in the diagram. The status of each link is independent of the status of the other links. What is the probability that there is a functioning set of links connecting computer B to computer C? B Uptime = 85% A Uptime = 93% Uptime = 96% C P = (0.5*0.93) + (0.5*0.85*0.96) = 0.873 3. Consider drawing one card from a standard 52-card deck. Define four events as follows: * +, * +, * +, and * +. Prove which pairs of these events are independent, and which are dependent. (A, C), (A, D), (B, C) and (B, D) are dependent and all others are independent 4. An urn contains 7 red balls and 5 green balls. Two balls are drawn at random from the urn. What is the probability of getting two balls of the same color if: (a) the first ball is replaced before the second ball is drawn. (b) the first ball is not replaced before the second ball is drawn. Module 11 Written Assignment - Solutions 2 a) Number of balls = 12 1st turn get red ball; probability = 7/12 2nd turn = 7/12 Probability of getting red in 2 turns = For green balls = Therefore probability with replacement = 0.340 + 0.173 = 0.513 b) Probability of getting red balls = Probability of getting blue balls = Probability without replacement = 5. Communication Channel Reliability: Consider a communication channel over which a 0 or a 1 must be transmitted. Suppose that the probability that the bit to be sent is a 0 is 0.5, and the probability that a 1 is to be sent is also 0.5. Also suppose that due to noise the probability that a 0 is changed to a 1 during transmission is 0.06 and the probability that a 1 is changed to a 0 is 0.03. Suppose that a 0 is received. What is the probability that a 0 was sent? A - event that 0 is received B - event that 0 is sent P(BIA) = ( ) ( ) P(A) = (0.5*0.03) + (0.5*0.94) = 0.485 ( ( ) ) Module 11 Written Assignment - Solutions 1. A fair die is rolled five times. a. What is the probability of getting five 6's in a row? b. What is the probability of getting all even numbers? a) If the dice is rolled once the probability will be chance it lands on 6. In 5 rolls be ( ) b) We have 3 even numbers. Therefore the probability will be . In 5 rolls it become ( ) 2. Network Reliability: In the network shown below, the links between computers can be either up or down. The reliability of each link is as shown in the diagram. The status of each link is independent of the status of the other links. What is the probability that there is a functioning set of links connecting computer B to computer C? A B Uptime = 85% Uptime = 93% Uptime = 96% C P= (0.5*0.93) + (0.5*0.85*0.96) = 0.873 3. Consider drawing one card from a standard 52-card deck. Define four events as follows: A={Hearts } , B={ cards } , C={ Aces } , and D={Queenof Hearts ,Queen of Spades } . Prove which pairs of these events are independent, and which are dependent. A - number of outcomes of getting hearts B - number of outcomes of getting black cards C - number of outcomes of getting aces D - number of outcomes of getting queen of hearts and queen of spades P(A)=13/52; P(B) = 26/52=1/2; P(B) = 4/52 = 1/13; P(D) = 2/52 = 1/26 Two vents are independent if 1 Module 11 Written Assignment - Solutions P ( A B ) =P ( A )P ( B ) ; P ( A B ) =0 ; P ( A C )= 2 1 1 1 1 ; P ( A D )= ; P ( B C )= ; P ( B D ) = ; P ( C 52 52 26 52 Therefore, (A, C), (A, D), (B, C), (B, D)are dependent and all others independent 4. An urn contains 7 red balls and 5 green balls. Two balls are drawn at random from the urn. What is the probability of getting two balls of the same color if: (a) the first ball is replaced before the second ball is drawn. (b) the first ball is not replaced before the second ball is drawn. 5. Communication Channel Reliability: Consider a communication channel over which a 0 or a 1 must be transmitted. Suppose that the probability that the bit to be sent is a 0 is 0.5, and the probability that a 1 is to be sent is also 0.5. Also suppose that due to noise the probability that a 0 is changed to a 1 during transmission is 0.06 and the probability that a 1 is changed to a 0 is 0.03. Suppose that a 0 is received. What is the probability that a 0 was sent