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
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