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Alice has an unbiased 5-sided die and 5 different coins with her. The probabilities of obtaining a head on tosses of these coins are 16, 26, 36, 46 and 56 respectively. She likes to observe patterns in subsequent tosses of these coins. Alice performs the following experiment. She rolls the 5-sided die and if the ith side turns up, she chooses the ith coin and starts tossing this coin repeatedly. 8 510 5 (a) What is the expected number of tosses required for obtaining 6 consecutive heads given that the side 1 turned up during the roll of the die? (b) What is the expected number of tosses required for obtaining 6 consecutive heads while performing this random experiment? (c) What is the probability that in the first n tosses, she obtains n consecutive heads? (d) In the first n-tosses, she obtains n consecutive heads. Given this outcome, calculate the probability that the ith side turned up during the roll of the die (the closed form expression for arbitrary i). How does these probabilities behave as n → ∞?

Alice has an unbiased 5-sided die and 5 different coins with her. The probabilities of obtaining a head on tosses of these coins are 16, 26, 36, 46 and 56 respectively. She likes to observe patterns in subsequent tosses of these coins. Alice performs the following experiment. She rolls the 5-sided die and if the ith side turns up, she chooses the ith coin and starts tossing this coin repeatedly. 8 510 5 (a) What is the expected number of tosses required for obtaining 6 consecutive heads given that the side 1 turned up during the roll of the die? (b) What is the expected number of tosses required for obtaining 6 consecutive heads while performing this random experiment? (c) What is the probability that in the first n tosses, she obtains n consecutive heads? (d) In the first n-tosses, she obtains n consecutive heads. Given this outcome, calculate the probability that the ith side turned up during the roll of the die (the closed form expression for arbitrary i). How does these probabilities behave as n → ∞?

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Counting And Probability
Section9.3: Binomial Probability
Problem 1E: A binomial experiment is UDC in which there are exactly __________ outcome. One outcome is called...
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Alice has an unbiased 5-sided die and 5 different coins with her. The probabilities of obtaining a head on tosses of these coins are 16 , 26 , 36 , 46 and 56 respectively. She likes to observe patterns in subsequent tosses of these coins. Alice performs the folowing experiment. She rolls the 5-sided die and if the ith side turns up, she chooses the ith coin and starts tossing this coin repeatedly. 8 510 5 (a) What is the expected number of tosses required for obtaining 6 consecutive heads given that the side 1 turned up during the roll of the die? (b) What is the expected number of tosses required for obtaining 6 consecutive heads while performing this random experiment? (c) What is the probability that in the first n tosses, she obtains n consecutive heads? (d) In the first n-tosses, she obtains n consecutive heads. Given this outcome, calculate the probability that the ith side turned up during the roll of the die (the closed form expression for arbitrary i). How does these probabilities behave as n → 0?
Transcribed Image Text:Alice has an unbiased 5-sided die and 5 different coins with her. The probabilities of obtaining a head on tosses of these coins are 16 , 26 , 36 , 46 and 56 respectively. She likes to observe patterns in subsequent tosses of these coins. Alice performs the folowing experiment. She rolls the 5-sided die and if the ith side turns up, she chooses the ith coin and starts tossing this coin repeatedly. 8 510 5 (a) What is the expected number of tosses required for obtaining 6 consecutive heads given that the side 1 turned up during the roll of the die? (b) What is the expected number of tosses required for obtaining 6 consecutive heads while performing this random experiment? (c) What is the probability that in the first n tosses, she obtains n consecutive heads? (d) In the first n-tosses, she obtains n consecutive heads. Given this outcome, calculate the probability that the ith side turned up during the roll of the die (the closed form expression for arbitrary i). How does these probabilities behave as n → 0?
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