I have an unfair coin for which P(H) = p, where 0 < p < 1....
I have an unfair coin for which P(H) = p, where 0 < p < 1. I toss the coin repeatedly until I observe heads for the first time. Let Y, be the total number of coin tosses. Find the distribution of Y. Hint: the range of Y is {1,2,3...}.
Homework Answers
Tossing an unfair coin with P(H) = 0.6 and P(T) = 0.4. The coin is tossed...
Tossing an unfair coin with P(H) = 0.6 and P(T) = 0.4. The coin is tossed 10 times (each toss is independent from others) and in any turn it shows heads, it is tossed again. We want to count the cases where the coin is tossed twice and the second toss, too, is head. For example, H T T T T T T T H T H T In this case, the count will be 1. Only the first turn...
Suppose we toss a coin (with P(H) p and P(T) 1-p-q) infinitely many times. Let Yi be the waiting time for the first head so (i-n)- (the first head occurs on the n-th toss) and Xn be the number of hea...
Suppose we toss a coin (with P(H) p and P(T) 1-p-q) infinitely many times. Let Yi be the waiting time for the first head so (i-n)- (the first head occurs on the n-th toss) and Xn be the number of heads after n-tosses so (X·= k)-(there are k heads after n tosses of the coin). (a) Compute the P(Y> n) (b) Prove using the formula P(AnB) P(B) (c) What is the physical meaning of the formula you just proved?
Suppose...
35. You and I play the following game: I toss a coin repeatedly. The coin is...
35. You and I play the following game: I toss a coin repeatedly. The coin is unfair and P(H) = p. The game ends the first time that two consecutive heads (HH) or two consec- utive tails (TT) are observed. I win if (HH) is observed and you win if (TT) is observed. Given that I won the game, find the probability that the first coin toss resulted in heads?
A defective coin minting machine produces coins whose probability of heads is a random variable P with PDF peP, p [0,1], otherwise fp(p) A coin produced by this machine is selected and tossed repeate...
A defective coin minting machine produces coins whose probability of heads is a random variable P with PDF peP, p [0,1], otherwise fp(p) A coin produced by this machine is selected and tossed repeatedly, with successive tosses assumed independent. (a) Find the probability that a coin toss results in heads. (b) Given that a coin toss resulted in heads, find the conditional PDF of P (c) Given that a first coin toss resulted in heads, find the conditional probability of...
h-0 h-0 h=0 or otherwise. (Oxford 1982M) 9. The probability of obtaining a head when a...
h-0 h-0 h=0 or otherwise. (Oxford 1982M) 9. The probability of obtaining a head when a certain coin is tossed is p. The coin is tossed repeatedly until n heads occur in a row. Let X be the total number of tosses required for this to happen. Find the expected value of X.
The next four questions (5 to 8) refer to the following: An unfair coin is tossed...
The next four questions (5 to 8) refer to the following: An unfair coin is tossed three times. For each toss, the probability that the coin comes up heads is 0.6 and the probability that the coin comes up tails is 0.4. If we let X be the number of coin tosses that come up heads, observe that the possible values of Xare 0, 1, 2, and 3. Find the probability distribution of X. Hint: the problem can be solved...
One application of an absolute value inequality is the concept of the unfair coin. If a coin is tossed 100 times, w...
One application of an absolute value inequality is the concept of the unfair coin. If a coin is tossed 100 times, we would expect approximately 50 of the tosses to be heads; however this is rarely the case.1. Toss a coin 100 times to test this hypothesis. Record the number of times the coin is heads and the number of times the coin is tails on the lines below. You may want to ask someone to tally the results of...
An experiment consists of tossing an unfair coin (49% chance of landing on heads) a specified...
An experiment consists of tossing an unfair coin (49% chance of landing on heads) a specified number of times and recording the outcomes. (a) What is the probability that the first head will occur on the second trial? (Use 4 decimal places.) Does this probability change if we toss the coin three times? What if we toss the coin four times? The probability changes if we toss the coin four times, but does not change if we toss the coin...
A coin with probability p of heads is tossed until the first head occurs. It is...
A coin with probability p of heads is tossed until the first head occurs. It is then tossed again until the first tail occurs. Let X be the total number of tosses required. (i) Find the distribution function of X. (ii) Find the mean and variance of X
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There are two steps in the description of this problem. First,
toss the coin until a head appears. Then, toss the coin until a
tail appears.
It is NOT "toss a coin until a head appears" problem.
=======================================================================================================================================================================================================================================================================================
A coin with probability p of heads is tossed until the first head occurs. It is then tossed again until the first tail occurs. Let X be the total number of tosses required (i) Find the distribution function of X (ii)...
Suppose we toss a coin (with P(H) p and P(T) 1-p-q) infinitely many times. Let Yi be the waiting time for the first head so (i-n)- (the first head occurs on the n-th toss) and Xn be the number of heads after n-tosses so (X·= k)-(there are k heads after n tosses of the coin). (a) Compute the P(Y> n) (b) Prove using the formula P(AnB) P(B) (c) What is the physical meaning of the formula you just proved?
Suppose...
35. You and I play the following game: I toss a coin repeatedly. The coin is unfair and P(H) = p. The game ends the first time that two consecutive heads (HH) or two consec- utive tails (TT) are observed. I win if (HH) is observed and you win if (TT) is observed. Given that I won the game, find the probability that the first coin toss resulted in heads?
A defective coin minting machine produces coins whose probability of heads is a random variable P with PDF peP, p [0,1], otherwise fp(p) A coin produced by this machine is selected and tossed repeatedly, with successive tosses assumed independent. (a) Find the probability that a coin toss results in heads. (b) Given that a coin toss resulted in heads, find the conditional PDF of P (c) Given that a first coin toss resulted in heads, find the conditional probability of...
h-0 h-0 h=0 or otherwise. (Oxford 1982M) 9. The probability of obtaining a head when a certain coin is tossed is p. The coin is tossed repeatedly until n heads occur in a row. Let X be the total number of tosses required for this to happen. Find the expected value of X.
The next four questions (5 to 8) refer to the following: An unfair coin is tossed three times. For each toss, the probability that the coin comes up heads is 0.6 and the probability that the coin comes up tails is 0.4. If we let X be the number of coin tosses that come up heads, observe that the possible values of Xare 0, 1, 2, and 3. Find the probability distribution of X. Hint: the problem can be solved...
A coin with probability p of heads is tossed until the first head occurs. It is then tossed again until the first tail occurs. Let X be the total number of tosses required. (i) Find the distribution function of X. (ii) Find the mean and variance of X
=======================================================================================================================================================================================================================================================================================
There are two steps in the description of this problem. First,
toss the coin until a head appears. Then, toss the coin until a
tail appears.
It is NOT "toss a coin until a head appears" problem.
=======================================================================================================================================================================================================================================================================================
A coin with probability p of heads is tossed until the first head occurs. It is then tossed again until the first tail occurs. Let X be the total number of tosses required (i) Find the distribution function of X (ii)...
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