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 flip a fair coin n times. We say that the sequence is balanced when there are equal number of heads and tails...
Please show ALL STEPS. NEAT HANDWRITING ONLY PLEASE Thank You 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 areHTHHTHTTHH, 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...
Flip a coin 10 times and record the observed number of heads and tails. For example, with 10 flips one might get 6 heads and 4 tails. Now, flip the coin another 20 times (so 30 times in total) and again, record the observed number of heads and tails. Finally, flip the coin another 70 times (so 100 times in total) and record your results again. We would expect that the distribution of heads and tails to be 50/50. How...
Suppose you flip a fair coin repeatedly until you see a Heads followed by another Heads or a Tails followed by another Tails (i.e. until you see the pattern HH or TT). (a)What is the expected number of flips you need to make? (b)Suppose you repeat the above with a weighted coin that has probability of landing Heads equal to p.Show that the expected number of flips you need is 2+p(1−p)/1−p(1−p)
Suppose we flip a coin three times, thereby forming a sequence of heads and tails. Form a random vector by mapping each outcome in the sequence to 0 if a head occurs or to 1 if a tail occurs. (a) How many realizations of the vector may be generated? List them. (b) Are the realizations independent of one another?
2. We flip a fair coin 5 times. Let A be the event that at least one T was flipped immediately after an H (i.e. the combination HT appears at least once in your sequence of flips). Use a Markov chain to compute P(A). Hint: Try using the following three states for your Markov chain: State 0: HT has not appeared yet and cannot appear in the next flip; State 1: HT has not appeared yet, but could appear in...
We flip a fair coin 5 times. Let A be the event that at least one T was flipped immediately after an H (i.e. the combination HT appears at least once in your sequence of flips). Use a Markov chain to compute P(A). Hint: Try using the following three states for your Markov chain: State 0: HT has not appeared yet and cannot appear in the next flip; State 1: HT has not appeared yet, but could appear in the...
Exercise 1.16. We flip a fair coin five times. For every heads you pay me $1 and for every tails I pay you $1. Let X denote my net winnings at the end of five flips. Find the possible values and the probability mass function of X.
We flip a fair coin 10 times. What is the probability that there are at least 4 heads out of the 10 flips?
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...
You toss a penny and observe whether it lands heads up or tails up. Suppose the penny is fair, i.e., the probability of heads is 1/2 and the probability of tails is y. This means every occurrence of a head must be balanced by a tail in one of the next two or three tosses. if I flip the coin many, many times, the proportion of heads will be approximately %, and this proportion will tend to get closer and...