Question

A coin is tossed repeatedly until two successive heads appear. Find the mean number of tosses required. Hint: Let $X_{n} $ be the cumulative number of successive heads. The state space is $0,1,2$, and the transition probability matrix is $$ P=1\left(\begin{array}{ccc} 0 & 1 & 2 0 & \frac{1}{2} & \frac{1}{2} &0 \frac{1}{2} & 0& \frac{1}{2} \ 2 & 0& 0 & 1 \end{array}\right. $$ Determine the mean time to reach state 2 starting from state 0 by invoking a first step analysis. SP.SS.379

Answer 1

To find the mean number of tosses required to reach two successive heads starting from no heads, we can set up a system of equations using the first step analysis. Solving this system of equations will give us the mean number of tosses required.

Step-by-step explanation:

First, we need to determine the transition probabilities for each state. From the given transition probability matrix, we can see that the probability of transitioning from state 0 to state 1 is 1, the probability of transitioning from state 0 to state 2 is 0, and the probability of transitioning from state 1 to state 0 is 0.5, from state 1 to state 1 is 0.5, and from state 1 to state 2 is 0. Finally, the probability of transitioning from state 2 to state 0 is 1.

To find the mean number of tosses required to reach state 2 starting from state 0, we can set up a system of equations using the first step analysis. Let T_0 be the mean number of tosses required to reach state 2 starting from state 0, T_1 be the mean number of tosses required to reach state 2 starting from state 1, and T_2 be the mean number of tosses required to reach state 2 starting from state 2.

• From state 0, the expected number of tosses to reach state 2 is 1 + the expected number of tosses to reach state 2 from state 1, which is T_1. The probability of transitioning from state 0 to state 1 is 1, so the equation is T_0 = 1 + T_1.
• From state 1, the expected number of tosses to reach state 2 is 1 + the expected number of tosses to reach state 2 from state 0, which is T_0, multiplied by the probability of transitioning from state 1 to state 0, which is 0.5, plus the expected number of tosses to reach state 2 from state 2, which is T_2, multiplied by the probability of transitioning from state 1 to state 2, which is 0.5. The equation is T_1 = 1 + 0.5*T_0 + 0.5*T_2.
• From state 2, the expected number of tosses to reach state 2 is 0. The equation is T_2 = 0.

Solving this system of equations, we can find the mean number of tosses required to reach state 2 starting from state 0.