Morse code uses \"dots\" and \"dashes,\" which are known to occur in the proport

Question

Morse code uses "dots" and "dashes," which are known to occur in the proportion of 4:3. This means that for any given symbol, P(dot sent) = 4/7 and P(dash sent) = 3/7. Unfortunately, when coded messages are sent, there are sometimes errors in transmission. Suppose there is interference on the transmission line, and with probability 1/10 a dot is mistakenly received as a dash, and vice versa.

a) What is the probability that the received symbol is a dot?

b) Given that we received a dot, what is the probability that it actually is a dot?

Explanation / Answer

Ans:

Given that

P(dot sent) = 4/7

P(dash sent) = 3/7

P(dash/dot sent)=1/10

P(dot/dash sent)=1/10

a)

P(dot)=P(dot/dash sent)*P(dash sent)+P(dot/dot sent)*P(dot sent)

=(1/10)*(3/7)+(9/10)*(4/7)

=39/70=0.557

b)

P(dot sent/dot)=P(dot/dot sent)*P(dot sent)/P(dot)

=(9/10)*(4/7)/(39/70)

=36/39

=0.923

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