The PMF for the result of any one roll of a three sided die with faces numbered

Question

The PMF for the result of any one roll of a three sided die with faces numbered 1,2 and 3 is Px(x)= Consider a sequence of six independent rolls of this die, and let X_i be the random variable corresponding to the result of the the roll. What is the probability that exactly two of the rolls have result equal to 2? What is the probability that the first roll is 1, given that exactly three of the six rolls have result of 1? We are told that exactly three of the rolls resulted in 1 and exactly three resulted in 2. Given this information, what is the probability that the sequence of rolls is 121212? 1

Explanation / Answer

1) WE HAD 6 ROLLS

PROBABILITY OF GETTING 2 ROLLS EQUAL TO 2

= 6C2*(0.25)^2*(0.75)^4 = 0.296

2)3 ROLLS ARE ONE

OTHER 3 ROLLS ARE EITHER 2 OR 3

WE NEED THE 1ST TO BE

SO TOTAL OPTIONS FOR REST SPACES = 3*3*2*2*2 = 54

TOTAL NUMBER WITHOUT 1ST BEING 1ST = 54*3 = 162

PROBABILITY = = 54/162 = 1/3 = 0.33

C)THREE ARE 1 AND 3 ARE TWO

TOTAL SPACES WHICH CAN BE FILLED WITH THIS CONDITION = 2*2*2*1*1*1 = 8

WE NEED THE PROBABILITY = 121212 WHICH IS ONLY 1 CASE

PROBABILITY = 1/8

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