Answer:
a. E(X) = 54.4
b. E(X) = 2.5
c. P(Y=2) = .0116
Step-by-step explanation:
a.
E(X) = np = .40 probability * 136 trials = 54.4 blocked transmissions
To get the expected value, we simply multiply probability times number of trials. You can look at it in simple terms by thinking if there's a 50% chance of flipping heads and you flip a coin twice, in an ideal world you will have .5*2 = 1 head.
b.
i. Let X represent the number of suspicious transmissions reviewed until finding the first blocked one. We will use a geometric distribution to model the "first" transmission. Whenever we're looking for the "first" time something happens, we use geometric.
ii. E(X) = 1/p , according to the geometric model.
= 1/.4 = 2.5.
We expect that the specialist will review 2.5 suspicious transactions <em>on average </em>before finding the first transmission that will be blocked.
c.
i. Let Y represent the exact number of blocked transmissions out of 10. We will use a binomial distribution to model the "fixed" number of transmissions. Whenever we're looking for a "fixed" number of times something happens, we use binomial.
ii. P(Y=k) = (n choose k)(p^k)(q^n-k)
P(Y=2) = (¹⁰₂)(.4^2)(.6^10-2)
= 45 (.4^2)(.6^10-2) = .0016
As for calculator notation, the n choose k can be accessed on a TI-84 via MATH -> PRB -> nCr. It looks like 10 nCr 2 on the display.
Hence the probability that two transactions out of ten will be blocked is .0016 by the binomial model.
(subtract 22/4 from each side)
(divide each side by 2)