Question

Suppose we toss a fair coin three times. Consider the events
*A*: we toss three heads, *B*: we toss at least one
head, and *C*: we toss at least two tails.

P(A) = 12.5

P(B) = .875

P(C) = .50

What is P(A ∩ B), P(A ∩ C) and P(B ∩ C)?

If you can show steps, that'd be great. I'm not fully sure what the difference between ∩ and ∪ is (sorry I can't make the ∪ bigger).

Answer #1

Suppose you toss a fair coin three times. Which of the following
events are independent? Give mathematical justification for your
answer.
A=
{“heads on first toss”}; B=
{“an odd number of heads”}.
A=
{“no tails in the first two tosses”}; B=
{“no heads in the second and third
toss”}.

A fair coin is tossed three times. What is the probability
that:
a. We get at least 1 tail
b. The second toss is a tail
c. We get no tails.
d. We get exactly one head.
e. You get more tails than heads.

a) Suppose we toss a fair coin 20 times. What is the probability
of getting between 9 and 11 heads?
b) Suppose we toss a fair coin 200 times. What is the
probability of getting between 90 and 110 heads?

You toss a fair coin four times. The probability of two heads
and two tails is

Toss a fair coin for three times and let X be the number of
heads.
(a) (4 points) Write down the pmf of X. (hint: first list all
the possible values that X can take, then calculate the probability
for X taking each value.)
(b) (4 points) Write down the cdf of X.
(c) (2 points) What is the probability that at least 2 heads
show up?

Suppose you toss a coin 100 times. Should you expect to get exactly
50 heads? Why or why not?
A. Yes, because the number of tosses is even, so if the coin
is fair, half of the results should be heads.
B. No, because the chance of heads or tails is the same, the
chance of any number of heads is the same.
C. No, there will be small deviations by chance, but if the
coin is fair, the result...

Suppose we toss a fair coin twice. Let X = the number of heads,
and Y = the number of tails. X and Y are clearly not
independent.
a. Show that X and Y are not independent. (Hint: Consider the
events “X=2” and “Y=2”)
b. Show that E(XY) is not equal to E(X)E(Y). (You’ll need to
derive the pmf for XY in order to calculate E(XY). Write down the
sample space! Think about what the support of XY is and...

Suppose a fair coin is flipped three times.
A). What is the probability that the second flip is heads?
B). What is the probability that there is at least two
tails?
C). What is the probability that there is at most two heads?

Suppose we toss a fair coin n = 1 million times and write down
the outcomes: it gives a Heads-and-Tails-sequence of length n. Then
we call an integer i unique, if the i, i + 1, i + 2, i + 3, . . . ,
i + 18th elements of the sequence are all Heads. That is, we have a
block of 19 consecutive Heads starting with the ith element of the
sequence. Let Y denote the number of...

Alan tosses a coin 20 times. Bob pays Alan $1 if the first toss
falls heads, $2 if the first toss falls tails and the second heads,
$4 if the first two tosses both fall tails and the third heads, $8
if the first three tosses fall tails and the fourth heads, etc. If
the game is to be fair, how much should Alan pay Bob for the right
to play the game?

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