Give the probability distribution for the indicated random variable.
A fair coin is tossed three times, and we count the number of "runs" that occur. A "run" is defined as a sequence of one or more of the same letter (e.g., HTH is 3 runs, TTH is 2 runs, and TTT is 1 run).

A) $\begin{array} { l }
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 6 } &\frac { 2 } { 3 } & \frac{ 1 } { 6 } \
\end{array}$
B) $\begin{array} { l }
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 3 } &\frac { 1 } { 3 } & \frac{ 1 } {3 } \
\end{array}$

C) $\begin{array} { l }
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 9 } &\frac { 7 } { 9 } & \frac{ 1 } { 9 } \
\end{array}$

D) $\begin{array} { l }
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } {8 } &\frac { 3 } { 4 } & \frac{ 1 } { 8 } \
\end{array}$

E) $\begin{array} { l }
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 4 } &\frac { 1 } { 2 } & \frac{ 1 } { 4 } \
\end{array}$

Question

Give the probability distribution for the indicated random variable.  
A fair coin is tossed three times, and we count the number of "runs" that occur. A "run" is defined as a sequence of one or more of the same letter (e.g., HTH is 3 runs, TTH is 2 runs, and TTT is 1 run).
 
A) $\begin{array} { l } 
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 6 } &\frac { 2 } { 3 } & \frac{ 1 } { 6 }  \
\end{array}$  
B)   $\begin{array} { l } 
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 3 } &\frac { 1 } { 3 } & \frac{ 1 } {3 }  \
\end{array}$
 
C)   $\begin{array} { l } 
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 9 } &\frac { 7 } { 9 } & \frac{ 1 } { 9 }  \
\end{array}$
 
D)   $\begin{array} { l } 
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } {8 } &\frac { 3 } { 4 } & \frac{ 1 } { 8 }  \
\end{array}$
 
E)   $\begin{array} { l } 
x & 1 & 2 & 3 \
P ( X = x ) &\frac { 1 } { 4 } &\frac { 1 } { 2 } & \frac{ 1 } { 4 }  \
\end{array}$

Quizplus · Accepted Answer

There are 8 possible outcomes when a fair coin is tossed three times: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. The number of runs for each outcome is: 1 run (HHH, TTT), 2 runs (HHT, HTT, THH, TTH), and 3 runs (HTH, THT). The probabilities are calculated as follows: P(X=1) = 2/8 = 1/4, P(X=2) = 4/8 = 1/2, P(X=3) = 2/8 = 1/4.

Give the Probability Distribution for the Indicated Random Variable x123P(X=x)162316\begin{array} { l } x & 1 & 2 & 3 \\P ( X = x ) &\frac { 1 } { 6 } &\frac { 2 } { 3 } & \frac{ 1 } { 6 } \\\end{array}xP(X=x)​161​​232​​361​​

Give the Probability Distribution for the Indicated Random Variable $\begin{array} { l } x & 1 & 2 & 3 \\P ( X = x ) &\frac { 1 } { 6 } &\frac { 2 } { 3 } & \frac{ 1 } { 6 } \\\end{array}$