Let H and T denote and outcome of getting a head, or a tail, respectively.
Consider the experiment of tossing three coins.
The sample space will be defined as,
[tex]\begin{gathered} S=\mleft\lbrace\text{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\mright\rbrace \\ n(S)=8 \end{gathered}[/tex]
As per the given options, X is assumed to be the random variable representing the number of heads obtained in a particular outcome of the experiment.
Consider that the probability of any favourable event (F) is given by,
[tex]P(F)=\frac{n(F)}{n(S)}[/tex]
Consider the event of getting no head.
[tex]\begin{gathered} F\colon\text{ no heads}(X=0) \\ F=\mleft\lbrace\text{TTT}\mright\rbrace \\ n(F)=1 \end{gathered}[/tex]
The corresponding probability is given by,
[tex]P(X=1)=\frac{3}{8}[/tex]
Consider the event of getting a head.
[tex]\begin{gathered} F\colon\text{ one heads}(X=1) \\ F=\mleft\lbrace\text{HTT, THT, TTH}\mright\rbrace \\ n(F)=3 \end{gathered}[/tex]
The corresponding probability is given by,
[tex]P(X=0)=\frac{1}{8}[/tex]
Thus, the probability of getting a head is,
[tex]\frac{3}{8}[/tex]
Consider the event of getting two head.
[tex]\begin{gathered} F\colon\text{ two heads}(X=2) \\ F=\mleft\lbrace\text{HHT, HTH, THH}\mright\rbrace \\ n(F)=3 \end{gathered}[/tex]
The corresponding probability is given by,
[tex]P(X=2)=\frac{3}{8}[/tex]
Consider the event of getting three head.
[tex]\begin{gathered} F\colon\text{ three heads}(X=3) \\ F=\mleft\lbrace\text{HHH}\mright\rbrace \\ n(F)=1 \end{gathered}[/tex]
The corresponding probability is given by,
[tex]P(X=3)=\frac{1}{8}[/tex]
Use the values to create the probability distribution table as follows,
Therefore, the 3rd option denotes the correct probability distribution.
