Question

Compute the probability distribution of getting a head when three coins are tossed.

Answer 1

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,

`\begin{gathered} S=\mleft\lbrace\text{HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}\mright\rbrace \\ n(S)=8 \end{gathered}`

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,

`P(F)=(n(F))/(n(S))`

Consider the event of getting no head.

`\begin{gathered} F\colon\text{ no heads}(X=0) \\ F=\mleft\lbrace\text{TTT}\mright\rbrace \\ n(F)=1 \end{gathered}`

The corresponding probability is given by,

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

Consider the event of getting a head.

`\begin{gathered} F\colon\text{ one heads}(X=1) \\ F=\mleft\lbrace\text{HTT, THT, TTH}\mright\rbrace \\ n(F)=3 \end{gathered}`

The corresponding probability is given by,

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

Thus, the probability of getting a head is,

`(3)/(8)`

Consider the event of getting two head.

`\begin{gathered} F\colon\text{ two heads}(X=2) \\ F=\mleft\lbrace\text{HHT, HTH, THH}\mright\rbrace \\ n(F)=3 \end{gathered}`

The corresponding probability is given by,

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

Consider the event of getting three head.

`\begin{gathered} F\colon\text{ three heads}(X=3) \\ F=\mleft\lbrace\text{HHH}\mright\rbrace \\ n(F)=1 \end{gathered}`

The corresponding probability is given by,

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

Use the values to create the probability distribution table as follows,

Therefore, the 3rd option denotes the correct probability distribution.