Answered: "a) To find: The probability that Y=0,1,2,3 using..."

Brittney Lord

Answered question

2021-09-24

a) To find: The probability that

Y = 0, 1, 2, 3

using the binomial distribution.
Given information:

n = 50

p = 0.01

b) To find: The probability that

Y = 0, 1, 2, 3

using the Poisson approximation.

Willie

Skilled2021-09-25Added 95 answers

a) Calculation:
Number of successes in a fixed number of independent trials follows a binomial distribution.
So, Binomial probability at

k = 0, 1, 2, or 3

P (Y = 0) = (b e g \in {a r r a y} {c} 50 ∖ 0 e n d {a r r a y}) .0 {.01}^{0} . {(1 - 0.01)}^{5 - 0} a p r r \otimes 0.60

P (Y = 1) = (b e g \in {a r r a y} {c} 50 ∖ 1 e n d {a r r a y}) .0 {.01}^{1} . {(1 - 0.01)}^{5 - 1} . \approx 0.30

P (Y = 2) = (b e g \in {a r r a y} {c} 50 ∖ 2 e n d {a r r a y}) .0 {.01}^{2} . {(1 - 0.01)}^{5 - 2} \approx 0.07

P (Y = 3) = (b e g \in {a r r a y} {c} 50 ∖ 3 e n d {a r r a y}) .0 {.01}^{3} . {(1 - 0.01)}^{5 - 3} \approx 0.01

Add all the probabilities

P (Y = 0, 1, 2, or 3) = P (Y = 0) + P (Y = 1) + P (Y = 2) + P (Y = 3)

= 0.60 + 0.30 + 0.07 + 0.01

= 0.98

The probability is 0.98.
b) Here,

μ = n p = 50 (0.01) = 0.5

λ = μ = 0.5

Probability at

k = 0, 1, 2

P (Y = 0) = \frac{{0.5}^{0} e^{- 0.5}}{0!} \approx 0.60

P (Y = 1) = \frac{{0.5}^{1} e^{- 0.5}}{1!} \approx 0.30

P (Y = 2) = \frac{{0.5}^{2} e^{- 0.5}}{2!} \approx 0.07

P (Y = 3) = {0.5}^{}

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