Question

Suppose you have an experiment where you flip a coin three times. You then count the number of heads. a.)State the random variable. b.)Write the probability distribution for the number of heads.

Answer 1

Answer:

a) X=number of heads observed when flipped the coin 3 times

b) the probability distribution is

P(X=x) = 3/4 * (1/[(3-x)!*x!)])

or

P(X)=1/8 for x=0 and x=3 and P(X)=3/8 for x=1 and x=2

Step-by-step explanation:

the random variable will be X=number of heads observed when flipped the coin 3 times . Since the result from each flip is independent of the others , then X has a binomial probability distribution , such that

P(X=x)= n!/[(n-x)!*x!)*p^x * (1-p)^(n-x)

where

n= number of times the coin is flipped = 3

p= probability of getting heads in a flip of the coin = 1/2 (assuming that the coin is fair)

therefore

P(X=x)= 3!/[(3-x)!*x!)*(1/2)^(3-x) * (1/2)^x = 3!/[(3-x)!*x!) * (1/2)³ = 3/4 * (1/[(3-x)!*x!)])

P(X=x)= 3/4 * (1/[(3-x)!*x!)])   , for x=[0,1,2,3]

for x=0 and x=3 → P(X)=3/4 * (1/[3!*0!)]) = 1/8

for x=1 and x=2 → P(X)=3/4 * (1/[2!*1!)]) = 3/8

we can verify that is correct since the sum of all the probabilities from x=0 to x=3 is  1/8 +  3/8+ 3/8+ 1/8 = 1

Answer 2

Answer:

a. Number of heads

b.

x      p(x)

0       1/8

1         3/8

2       3/8

3        1/8

Step-by-step explanation:

a)

A coin is flipped three times and the number of heads are counted.

We are interested in counting heads so, a random variable X is the number of heads appears on a coin.

b)

The sample space for flipping a coin three times is

S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}

n(S)=8

The random variable X (number of heads) can take values 0,1,2 and 3 .

0 head={TTT}

P(0 heads)=P(X=0)=1/8

1 head={HTT,THT,TTH}

P(1 head)= P(X=1)=3/8

2 heads= {HHT,HTH,THH}

P(2 heads)=P(X=2)=3/8

3 heads={HHH}

P(3 heads)=1/8

The probability distribution for number of heads can be shown as

x      p(x)

0       1/8

1         3/8

2       3/8

3        1/8