Consider an experiment where you flip four coins. Identify the possible outcomes of this experiment. Calculate

Question

the conditional probability that all four flips will be heads. Then calculate the conditional probability of flipping four heads, given that the first two flips were heads. Now assign a random variable x, where the value of x is equal to the number of heads observed in the flips. Describe the probability distribution of x, and calculate the probability of observing three heads in four coin flips. (I don't even know where to start on this so if any statistics buffs could help me out I would really appreciate it)

Answer 1

Since each flip is independent of the prior flips, it is just the probability the first is heads, and the second is heads, and the third is heads and the fourth is heads.

P(all four heads) = 1/2 x 1/2 x 1/2 x 1/2 = 1/16

The probability that you have four heads, given the first two were heads, is just the probability that flips 3 and 4 are heads. Again, since the events are independent, it is the chance the third is heads and the fourth is heads.

P(four heads, given the first two are heads) = 1/2 x 1/2 = 1/4

For figuring the distribution, it is probably easiest to enumerate the 16 possibilities and count the number of heads:
HHHH = 4
HHHT = 3
HHTH = 3
HHTT = 2
HTHH = 3
HTHT = 2
HTTH = 2
HTTT = 1
THHH = 3
THHT = 2
THTH = 2
THTT = 1
TTHH = 2
TTHT = 1
TTTH = 1
TTTT = 0

Distribution:
0 = 1/16
1 = 4/16 = 1/4
2 = 6/16 = 3/8
3 = 4/16 = 1/4 <---
4 = 1/16

P (3 heads in 4 flips) = 1/4

Answer 2

There is no such thing as the conditional probability that all 4 coins are heads, because there is nothing observed. We say "the probability that all four coins are heads" (which is (1/2)x(1/2)x(1/2)x(1/2) = 1/16). We can ask the conditional probability that 4 heads are flipped given that we know the first two are heads. It is just the probability that the last two coin flips are both heads (1/2)x(1/2) = 1/4.

Answer 3

Assuming that the coin is unbiased.

P (4 heads) = 1/2 * 1/2 * 1/2 *1/2

Conditional probability of 4 heads given 2 heads is the same as flipping 2 heads in 2 flips,

ie 1/2*1/2

x ~ Bin (4, 0.5)

P (3 heads) = 4 * (0.5^4)

Answer 4

well, if order does not matter, there are 5 possible outcomes:
hhhh
t t t t
t t t h
t t hh
t hhh
there is a .5 probability that each flip will be h or t. .5^4=.0625 for all heads. I forget how to do the others but maybe that will help?

Answer 5

These are standard probability questions which I am sure, using the proper mathematics you can solve.....HOWEVER....What is the probability all four coins will land on their edges and stand straight up....a real question for real thinkers.

Answer 6

Do your own homework.
GO SHS