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DM-Probability -Q18

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Suppose E is the event that a randomly generated bit string of length four begins with a 1 and F is the event that this bit string contains an even number of 1s. if the 16 bit strings of length four are equally likely, which of the following statements is/are correct?

1. E and F are independent events.

2. $\dpi{100} P(E \cap F ) = 1/4$

(A). Only 1

(B). Only 2

(C). Both 1 and 2

(D). None

reshown Jul 10, 2020

+1 vote

There are 8 bit strings of length four that begin with a 1,
and 8bit strings of length four that contain an even number of 1s.

Since the number of bit strings of length 4 is 16,

Since E⋂F = {1111, 1100, 1010, 1001},
p(E⋂F) = 4/16=1/4.
We conclude that E and F are independent,
because p(E⋂F) =1/4 = (½) (½)= p(E) p(F)

answered Jul 11, 2020 by (5,820 points)
selected Jul 11, 2020