Question

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 should be close to 50 heads

D. Yes, because the chance of heads or tails is the same, the
number of heads and the number of tails should be the same.

Answer #1

Suppose you toss a fair coin 10,000 times. Should you expect to
get exactly 5000 heads? Why or why not? What does the law of
large numbers tell you about the results you are likely to get?
Choose the correct answer below.
1)Should you expect to get exactly 5000 heads? Why or why
not?
A)You shouldn't expect to get exactly 5000 heads, because you
cannot predict precisely how many heads will occur.
B.You shouldn't expect to get exactly 5000 heads,...

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”}.

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?

I toss a coin two times. X1 is the number of heads on the first
toss. X2 is the number of heads on the second toss.
Find the mean of X1.
Find the variance of X1.
Find the mean of X1 + X2. (This is the number of heads in 2
tosses.)
Find the variance of X1 + X2.
If you tossed 10 coins, how many heads would you expect? What is
the variance of the number of heads?

(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...

You toss a coin 192 times. Find the probability that exactly 96
of the 192 tosses will be heads.

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

(a) Use the central limit theorem to determine the probability
that if you toss a coin 50 times, you get fewer than 20 heads.
(b) A coin is continuously tossed until the heads come up 20th
time. Use the central limit theorem to estimate the probability
that more than 50 coin tosses are required to get the 20th
head.
(c) Compare your answers from parts (a) and (b). Why are they
close but not exactly equal?

Suppose you toss an unfair coin 8 times independently. The
probability of getting heads is 0.3. Denote the outcome to be 1 if
you get heads and 0 if you get tails.
1.Write down the sample space.
2. What is the probability of the event that you get a head or a
tail at least once?
3. If you get 8 same toss you will get x dollars, otherwise you
will lose one dollar. On average, how large should x...

If you flip a fair coin and get heads 10 times in a row, what is
the chance of getting tails on the next flip? Explain.

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