Dice coin probability

We roll the dice and then toss the coin as many times as the number that came up on the dice. What is the probability that the coin lands head at least once?

Final Answer:

p =  0.8359

Step-by-step explanation:

$$\begin{gathered} p_0=1/2=\frac{1}{2}=0.5 \\ {p}_1=p_0=\frac{1}{2}=0.5 \\ {p}_2=1-p_0^2=1-{\frac{1}{2}}^2=\frac{3}{4}=0.75 \\ {p}_3=1-p_0^3=1-{\frac{1}{2}}^3=\frac{7}{8}=0.875 \\ {p}_4=1-p_0^4=1-{\frac{1}{2}}^4=\frac{15}{16}=0.9375 \\ {p}_5=1-p_0^5=1-{\frac{1}{2}}^5=\frac{31}{32}\doteq 0.9688 \\ {p}_6=1-p_0^6=1-{\frac{1}{2}}^6=\frac{63}{64}\doteq 0.9844 \\ p=\frac{1}{6}\cdot (p_1+p_2+p_3+p_4+p_5+p_6)=\frac{1}{6}\cdot (0.5+0.75+0.875+0.9375+0.9688+0.9844)=\frac{107}{128}=0.8359 \end{gathered}$$

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