a) In order to complete the table for the probability distribution for the variable X:
X = the number of heads observed when you flip a coin three times,
b) we create a list of all the possible outcomes. We use the notation: H = heads, T = tails.
The possible outcomes after tossing the coin three times are:
[tex]HHH,HHT,HTH,THH,TTT,TTH,THT,HTT[/tex]
From the list above we see that we have 8 equiprobable events. Now, we classify the events according to the number of Heads included.
- Events with x = #H = 0: TTT
- Events with x = #H = 1: HTT, THT, TTH
- Events with x = #H = 2: HHT, HTH, THH
- Events with x = #H = 3: HHH
Using the data above we compute the probabilities for each x, we simply compute the quotient between the number of outcomes of each case by N = 8 (the total number of possible outcomes):
[tex]\begin{gathered} P\mleft(x=0\mright)=\frac{1}{8}=0.125 \\ P(x=1)=\frac{3}{8}=0.375 \\ P(x=2)=\frac{3}{8}=0.375 \\ P(x=3)=\frac{1}{8}=0.125 \end{gathered}[/tex]
c) Shape of the probability disribution. The probability distribution of x is: Symmetric.
d) Using the data above, we can compute the mean number of heads for the distribution:
[tex]\begin{gathered} E(x)=\sum ^3_{n=0}x_n\cdot P(x_n_{}) \\ E(x)=0\cdot P(x=0)+1\cdot P(x=1)+2\cdot P(x=2)+3\cdot P(x=3) \\ E(x)=0+1\cdot0.375+2\cdot0.375+3\cdot0.125 \\ E(x)=\frac{3}{2}=1.5 \end{gathered}[/tex]
So the mean number of heads for this distribution is 1.5
