Respuesta :
Answer:
If Tristan rolled the dice first the probability that Tristan wins is 0.474.
Step-by-step explanation:
The probability of an event E is computed using the formula:
[tex]P(E)=\frac{Favorable\ otucomes}{Total\ outcomes}[/tex]
Given:
Tristan and Iseult play a game where they roll a pair of dice alternatively until Tristan wins by rolling a sum 9 or Iseult wins by rolling a sum of 6.
The sample space of rolling a pair of dice consists of a total of 36 outcomes.
The favorable outcomes for Tristan winning is:
S (Tristan) = {(3, 6), (4, 5), (5, 4) and (6, 3)} = 4 outcomes
The favorable outcomes for Iseult winning is:
S (Iseult) = {(1, 5), (2, 4), (3, 3), (4, 2) and (5, 1)} = 5 outcomes
Compute the probability that Tristan wins as follows:
[tex]P(E)=\frac{Favorable\ otucomes}{Total\ outcomes}\\ P(Tristan\ wins)=\frac{4}{36}\\P(T)=\frac{1}{9} \\\approx0.1111[/tex]
Compute the probability that Iseult wins as follows:
[tex]P(E)=\frac{Favorable\ otucomes}{Total\ outcomes}\\ P(Iseult\ wins)=\frac{4}{36}\\P(I)=\frac{1}{9} \\\approx0.1111[/tex]
If Tristan plays first, then the probability that Tristan wins is:
= P(T) + P(T')P(I')P(T) + P(T')P(I')P(T')P(I')P(T)+...
=P(T) + [(1-P(T))(1-P(I))P(T)]+[(1-P(T))(1-P(I))(1-P(T))(1-P(I))P(T)]+...
[tex]=0.1111+(0.8889\times0.8611\times0.1111)+(0.8889\times0.8611\times0.8889\times0.8611\times0.1111))+...\\=0.1111[1+(0.8889\times0.8611)+(0.8889\times0.8611)^{2}+...]\\[/tex]This is an infinite geometric series.
The first term is, a = 0.1111 and the common ratio is, r = (0.8889×0.8611).
The sum of infinite geometric series is:
[tex]S_{\infty}=\frac{a}{1-r}\\ =\frac{0.1111}{1-(0.8889\times0.8611}\\ =0.47364\\\approx0.474[/tex]
Thus, the probability that Tristan wins if he rolled the die first is 0.474.
