Question

1) If you roll 4 fair dice, what is the probability that the sum of the face-up values is less than 22 OR all the face-up values are equal?

2) Billy and Cam are playing the following game: each player has a coin and decides whether to leave it as heads or tails before showdown (both player reveals their coin simultaneously). If both coins are heads, Billy wins $2. If both are tails, Billy wins $0.50. Otherwise, Cam wins $1. Find the optimal strategy for Billy.

3) A day on a planet X can be in 2 states: sunny or rainy. The climate of planet X is determined by the following pattern: every sunny day is followed by a sunny day with a probability of 2/3, and every rainy day is followed by a rainy day with a probability of 1/3.

(a) Find the transition matrix that represents the above change in weather pattern.

(b) Find a steady state vector for the above Markov system.

Answer 1

1) when 4 fair dice are rolled Total number of outcome = 69=1296 som af face value the face - up nalues are equal P(xuy) = P(x) + P(4) - P(xny) P(x<22)=1-P(x>,22) PC X7,22) = P(x=22) + P(x=23) +P(x=24) sum as 24 can be obtained of face up valle are (6, 6, 6, 6) l.e., obtaining som af 24 there is only I case (6, 6, 6, 6) Sum of 23 obtained when we have 3 six & Ane it can be obtained in uc, cases we have 3-six & so Sum of 22 obtained when & two 1 - four total ways PCX7, 22)= uc, +4 = 10 case 1296 - P(x222)=1-15 1296 P(x(22)=0.988 P(x<22)=1281 1296

"P(Y) = 6 1296 be que whes 4 dice are rolled there can chance that we can get (1,1,1,1),(2,2,2,2) (3, 3, 3, 3), (4,4,4,4),(5,5,5,5), 6, 6, 6, 6) ) so cases when all face up value same ; ply)= 1296 Plany) = 1 because in cases cases of 1296 X and Y (6, 6, 6, 6) is common Plxuy) = P(x222)+P(y)- P(x22204) 1281 1296 + 1896-1 1296 P(Xuy) -0.988 1286 19.96 sum af Probability that face up value is less than 22 ar the face up values are same (equal)