For the fair coin, we are given here that
P(heads) = P(tails) = 1/2 = 0.5
Therefore the expected number of total tosses required to get the third heads is computed using the expected value of the negative binomial distribution here as:
= n/p = 3/(1/2) = 6
As the expected number of total tosses required to get 3 heads
is 6, therefore total expected number of tails before the 3 heads
would be computed as:
= 6 - 3 = 3
Therefore 3 is the expected number of tails here.
Exercise 2. C What is the expected number of Tails until I get the third Heads...
What is the expected number of Tails until I get the third Heads in an infinite sequence of independent coin tosses with probability 1/2 for each?
What is the expected number of Tails until I get the third Heads in an infinite sequence of independent coin tosses with probability 1/2 for each?
17. A fair coin is tossed until either one Heads or four Tails are obtained. What is the expected number of tosses? [6 points]
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...
3. (PMF – 8 points) Consider a sequence of independent trials of fair coin tossing. Let X denote a random variable that indicates the number of coin tosses you tried until you get heads for the first time and let y denote a random variable that indicates the number of coin tosses you tried until you get tails for the first time. For example, X = 1 and Y = 2 if you get heads on the first try and...
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?
You have a biased coin where heads come up with probability 2/3
and tails come up with probability 1/3.
2. Assume that you flip the coin until you get three heads or one tail. (a) Draw the possibility tree. (b) What is the average number of flips? Use the possibility tree, and show your calculation.
2. Assume that you flip the coin until you get three heads or one tail. (a) Draw the possibility tree. (b) What is the average...
Q3. Suppose we toss a coin until we see a heads, and let X be the number of tosses. Recall that this is what we called the geometric distribution. Assume that it is a fair coin (equal probability of heads and tails). What is the p.m.f. of X? (I.e., for an integer i, what is P(X=i)? What is ?[X]? ({} this is a discrete variable that takes infinitely many values.)
Problem 2: Tails and (Heads or Tails?) Alice and Bob play a coin-tossing game. A fair coin (that is a coin with equal probability of 1. The coin lands 'tails-tails' (that is, a tails is immediately followed by a tails) for the first 2. The coin lands 'tails-heads (that is, a tails is immediately followed by a heads) for the landing heads and tails) is tossed repeatedly until one of the following happens time. In this case Alice wins. first...
4. A fair two-sided coin is tossed repeatedly. (a) Find the expected number of tails until the first head is flipped. (b) Find the probability that there are exactly 5 heads in the first 10 flips. (c) Use the central limit theorem/normal approximation to approximate the probability that in the first 100 flips, between 45 and 55 of the flips are heads.