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?
Homework Answers
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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?
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?
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...
TABLE II. Probability of Independent Events Occurring COMBINATIONS OBSERVED % EXPECTED OBSERVED 1% EXPECTED % 27500...
TABLE II. Probability of Independent Events Occurring COMBINATIONS OBSERVED % EXPECTED OBSERVED 1% EXPECTED % 27500 DEVIATION Heads-Heads 1754 10 25 60000 Heads-Tails or Tails-Heads 20 50 + 4 16000 Tails-Tails o 600/0 22.560 100% 10 40 25 100% -1 o 5000 Total 1000 PART III: Probability and Mendelian Genetics we can use the probability to predict the probability of given genetic traits appearing in the offspring of particular parents. Punnett squares can also be used to obtain these results....
4. [20 Points We first examine a sequence of rolls of a four-sided die at an the observed outcome Xi E {1,2,3,4}. At ea...
4. [20 Points We first examine a sequence of rolls of a four-sided die at an the observed outcome Xi E {1,2,3,4}. At each of these times, the casino can be in one of two states z E1, 2}. When z = 1 the casino uses a fair die, while when z = 2 the die is biased so that rolling a 1 is more simple hidden Markov model (HMM). We observe a "occasionally dishonest casino", where at time likely....
Can Dogs Understand Human Cues? EXPLORATION Dogs have been domesticated for about 14,000 years. In that...
Can Dogs Understand Human Cues? EXPLORATION Dogs have been domesticated for about 14,000 years. In that time, have they been able to develop an understanding of human gestures such as pointing or glancing? How about simi lar nonhuman cues? Researchers Udell, Giglio, and Wynne tested a small number of dogs in order to answer these questions. In this exploration, we wll first see whether dogs can understand human gestures as well as nonhuman gestures. To test this, the researchers positioned...
CAN SOMEONE HELP WITH THIS? Part 1: power() method We want to write a method that...
CAN SOMEONE HELP WITH THIS? Part 1: power() method We want to write a method that will enable us to calculate a number to a given power. In other words, we are writing a method to mimic what the Math.pow() method we have seen before does. To write this method use the following requirements: The method should return an integer. It should have two parameters x and y both integers. The name of this method will be power(). For the...
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,...
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...
TABLE II. Probability of Independent Events Occurring COMBINATIONS OBSERVED % EXPECTED OBSERVED 1% EXPECTED % 27500 DEVIATION Heads-Heads 1754 10 25 60000 Heads-Tails or Tails-Heads 20 50 + 4 16000 Tails-Tails o 600/0 22.560 100% 10 40 25 100% -1 o 5000 Total 1000 PART III: Probability and Mendelian Genetics we can use the probability to predict the probability of given genetic traits appearing in the offspring of particular parents. Punnett squares can also be used to obtain these results....
4. [20 Points We first examine a sequence of rolls of a four-sided die at an the observed outcome Xi E {1,2,3,4}. At each of these times, the casino can be in one of two states z E1, 2}. When z = 1 the casino uses a fair die, while when z = 2 the die is biased so that rolling a 1 is more simple hidden Markov model (HMM). We observe a "occasionally dishonest casino", where at time likely....
Can Dogs Understand Human Cues? EXPLORATION Dogs have been domesticated for about 14,000 years. In that time, have they been able to develop an understanding of human gestures such as pointing or glancing? How about simi lar nonhuman cues? Researchers Udell, Giglio, and Wynne tested a small number of dogs in order to answer these questions. In this exploration, we wll first see whether dogs can understand human gestures as well as nonhuman gestures. To test this, the researchers positioned...
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