Question

A fair coin is tossed for n times independently. (i) Suppose that n = 3. Given the appearance of successive heads, what is the conditional probability that successive tails never appear? (ii) Let X denote the probability that successive heads never appear. Find an explicit formula for X. (iii) Let Y denote the conditional probability that successive heads appear, given no successive heads are observed in the first n − 1 tosses. What is the limit of Y as n goes to infinity?

Answer #1

Given a fair coin, if the coin is flipped n times, what is the
probability that heads is only tossed on odd numbered tosses.
(tails could also be tossed on odd numbered tosses)

(a) A fair coin is tossed five times. Let E be the event that an
odd number of tails occurs, and let F be the event that the first
toss is tails. Are E and F independent?
(b) A fair coin is tossed twice. Let E be the event that the
first toss is heads, let F be the event that the second toss is
tails, and let G be the event that the tosses result in exactly one
heads...

A fair coin is tossed three times. Let X be the number of heads
among the first two tosses and Y be the number of heads among the
last two tosses. What is the joint probability mass function of X
and Y? What are the marginal probability mass function of X and Y
i.e. p_X (x)and p_Y (y)? Find E(X) and E(Y). What is Cov(X,Y) What
is Corr (X,Y) Are X and Y independent? Explain. Find the
conditional probability mass...

NOTE:KINDLY SOLVE PARTS D AND E.
A fair coin is tossed four times, and the random variable X is
the number of heads in the first three tosses and the random
variable Y is the number of heads in the last three tosses.
a) What is the joint probability mass function of X and Y ?
b) What are the marginal probability mass functions of X and Y
?
c) Are the random variables X and Y independent?
d) What...

A fair coin has been tossed four times. Let X be the number of
heads minus the number of tails (out of four tosses). Find the
probability mass function of X. Sketch the graph of the probability
mass function and the distribution function, Find E[X] and
Var(X).

Let p denote the probability that a particular coin will show
heads when randomly tossed. It is not necessarily true that the
coin is a “fair” coin wherein p=1/2. Find the a posteriori
probability density function f(p|TN ) where TN is the observed
number of heads n observed in N tosses of a coin. The a priori
density is p~U[0.2,0.8], i.e., uniform over this interval. Make
some plots of the a posteriori density.

Suppose that a fair coin is tossed 10 times.
(a) What is the sample space for this experiment?
(b) What is the probability of at least two heads?
(c) What is the probability that no two consecutive tosses come
up heads?

A balanced coin is tossed 3 times, and among the 3 coin tosses,
X heads show. Then the same balanced coin is tossed X additional
times, and among these X coin tosses, Y heads show.
a. Find the distribution for Y .
b. Find the expected value of Y .
c. Find the variance of Y .
d. Find the standard deviation of Y

A balanced coin is tossed 3 times, and among the 3 coin tosses,
X heads show. Then the same balanced coin is tossed X additional
times, and among these X coin tosses, Y heads show.
a. Find the distribution for Y .
b. Find the expected value of Y .
c. Find the variance of Y .
d. Find the standard deviation of Y

A biased coin is tossed ten times. Given that exactly four Heads
were obtained, what is the conditional probability that exactly two
Heads were obtained in the first five tosses?

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