Let us consider the following experiment: we roll two dice consecutively. The fi

Question

Let us consider the following experiment: we roll two dice consecutively. The first die is fair. If the first roll is a 1, 2, or 3, we roll the same die again. If the first roll yields a 4, 5, or 6, then we roll an unfair die with probabilities [0 0 0 1/3 1/3 1/3]. What is the probability of the outcome (1,6) (the first roll is a 1 and the second is a 6)? What is the probability of the outcome (6,1)? What is the probability of the outcome (6, 6)? In addition to calculating these probabilities analyticaly, please write a matlab script to simulate this experiment. Run 1000 trials of this experiment and estimate the same probabilities you calculated analytically.

Explanation / Answer

a) the probability of first roll 1 is 1/6 and for second roll probability of it rolled by first dice is 1/2 in this case probability of 1 is again 1/6 and in second dice probability of 1 is 0. so total probaility is 1/6*(1/2*1/6 + 1/2*0) = 1/72.
b) probability of 6,6 => probability of first 6 is 1/6 and probability of 6 by firstdice = 1/2*1/6 and probability of 6 by second dice= 1/2*1/3 = so total probability= 1/6*(1/2*1/6+ 1/2*1/3)= 1/24

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