A Question From Probability

Topic : Probability Word Problem

Question : In transmitting dot and dash signals, a communication system changes ¼ of the dots to dashes and 1/3 of the dashes to dots. If 40% of the signals transmitted are dots and 60% are dashes, what is the probability that a dot received was actually a transmitted dot?

Solution :
[a]Dot Signals transmitted = 40%
[b]Dash signals transmitted = 60%
Let A be the event of receiving a dot.
And B the event of a transmitted dot received as dot.
So the required probability P(BA). = P(A and B)/P(A).
P(A) = 3/4 X 0.40 + 1/3 X 0.60 = 0.30 + 0.20 = 0.50
P(A and B) = 3/4 X 0.40 = 0.30
So the required Probability = P(BA) = 0.30/0.50 = 3/5.