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The communication channel transmits a three-digit number M = 873, written in decimal digits.
Each digit is passed correctly with probability 0.62 and is converted to any other digit
with probability (1 - 0.62) / 9.
a) what is the probability that transmitting the number M will decrease it?
b) what is the maximum increase in the probability of a decrease in the number when adding a digit from the right?
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I will solve part (a), only.
The probability that transmitting the number M= 873 will increase it is this sum
P = P(first digit 8 is no change)*P(second digit 7 is no change)*P(third digit 3 increases) +
+ P(first digit 8 is no change)*P(second digit 7 increases) +
+ P(first digit 8 is increases) =
= 0.62*0.62*(6*((1-0.62)/9)) + 0.62*(2*((1-0.62)/9)) + 1*((1-0.62)/9) = 0.191959 (rounded). ANSWER
It requires some comments.
P(third digit 3 increases) = 6*((1-0.62)/9), since there are 6 possibilities to increase 3: they are 4,5,6,7,8,9.
P(second digit 7 increases) = 2*((1-0.62)/9), since there are 2 possibilities to increase 7: they are 8,9.
P(first digit 8 is increases) = 1*((1-0.62)/9), since there is only 1 possibility to increase 8: it is 9.
Solved.