When we toss a two-sided coin, we cannot predict the outcome of a single toss. Therefore,...
When we toss a two-sided coin, we cannot predict the outcome of a single toss. Therefore, we cannot predict the next term in the sequence HHTTHHTTHH…, where H represents the coin landing on heads and T represents the coin landing on tails.
- What aspect of the sequence is predictable if we toss the coin 5000 times?
- What aspect of the sequence is unpredictable, besides the result of the next toss, if we toss the coin 15 times?
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When we toss a two-sided coin, we cannot predict the outcome of
a single toss. Therefore,...
9.74. Suppose we toss a biased coin independently until we get two heads or two tails...
9.74. Suppose we toss a biased coin independently until we get two heads or two tails in total. The coin produces a head with probability p on any toss. 1. What is the sample space of this experiment? 2. What is the probability function? 3. What is the probability that the experiment stops with two heads?
A coin is tossed 23 times, and the sequence of heads and tails is the outcome....
A coin is tossed 23 times, and the sequence of heads and tails is the outcome. A statistical test is conducted for the following hypotheses. H,: The coin is a fair coin. H,: The chance of obtaining a head is three time as the chance of obtaining a tail. The critical region for the test is the event “more than k heads”. Here k is a positive integer. If we want the power of the test to be at least...
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...
Example 5.5. We roll a fair die then toss a coin the number of times shown...
Example 5.5. We roll a fair die then toss a coin the number of times shown on the die. What is the probability of the event A that all coin tosses result in heads? One could use the state space Ω = {(1, H), (1, T), (2, H, H), (2, T, T), (2, T, H), (2, H, T), . . . }. However, the outcomes are then not all equally likely. Instead, we continue the state space is Ω {1,...
Suppose we flip a fair coin n times. We say that the sequence is balanced when there are equal number of heads and tails...
Suppose we flip a fair coin n times. We say that the sequence is balanced when there are equal number of heads and tails. For example, if we flip the coin 10 times and the results are HT HHT HT T HH, then this sequence balanced 2 times, i.e. at position 2 and position 8 (after the second and eighth flips). In terms of n, what is the expected number of times the sequence is balanced within n flips?
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...
Question 2 Suppose you have a fair coin (a coin is considered fair if there is...
Question 2 Suppose you have a fair coin (a coin is considered fair if there is an equal probability of being heads or tails after a flip). In other words, each coin flip i follows an independent Bernoulli distribution X Ber(1/2). Define the random variable X, as: i if coin flip i results in heads 10 if coin flip i results in tails a. Suppose you flip the coin n = 10 times. Define the number of heads you observe...
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...
Problem 5 - Rare outcomes and data set size Here we will be concerned with a biased coin for whic...
Problem 5 - Rare outcomes and data set size Here we will be concerned with a biased coin for which outcome 1 has a very low probability, i.e 0 < θι < 6o << 1. Assume our experiment consists of n independent tosses of this coin. 1. What is the probability po P(n1 0) that the outcome sequence contains no 1's? Write the answer as a function of θ| and n 2. What is the probability pi P(n1-1) that the...
The next three questions (5 to 8) refer to the following: An unfair coin is tossed...
The next three 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 X are 0, 1, 2, and 3. Find the probability distribution of X. Hint: the problem can be...
A coin is tossed 23 times, and the sequence of heads and tails is the outcome. A statistical test is conducted for the following hypotheses. H,: The coin is a fair coin. H,: The chance of obtaining a head is three time as the chance of obtaining a tail. The critical region for the test is the event “more than k heads”. Here k is a positive integer. If we want the power of the test to be at least...
Example 5.5. We roll a fair die then toss a coin the number of times shown on the die. What is the probability of the event A that all coin tosses result in heads? One could use the state space Ω = {(1, H), (1, T), (2, H, H), (2, T, T), (2, T, H), (2, H, T), . . . }. However, the outcomes are then not all equally likely. Instead, we continue the state space is Ω {1,...
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...
Question 2 Suppose you have a fair coin (a coin is considered fair if there is an equal probability of being heads or tails after a flip). In other words, each coin flip i follows an independent Bernoulli distribution X Ber(1/2). Define the random variable X, as: i if coin flip i results in heads 10 if coin flip i results in tails a. Suppose you flip the coin n = 10 times. Define the number of heads you observe...
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...
Problem 5 - Rare outcomes and data set size Here we will be concerned with a biased coin for which outcome 1 has a very low probability, i.e 0 < θι < 6o << 1. Assume our experiment consists of n independent tosses of this coin. 1. What is the probability po P(n1 0) that the outcome sequence contains no 1's? Write the answer as a function of θ| and n 2. What is the probability pi P(n1-1) that the...
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