Solved: "Interpretation and calculation for the probability of event..."

generals336

Answered question

2021-09-20

Interpretation and calculation for the probability of event

Y \geq 7

.
Given information:
Number of trials,

n = 10

Probability of success,

p = 55 % = 0.55

Y: The number of times that the light is red.

Answer & Explanation

Nicole Conner

Skilled2021-09-21Added 97 answers

Calculation:
The binomial probability indicates that
$P (X = k) = (b e g \in {a r r a y} {c} n ∖ k e n d {a r r a y}) \cdot p^{k} \cdot {(1 - p)}^{n - k}$
Rule for adding events that are mutually exclusive:
$P (A \cup B) = P (A or B) = P (A) + P (B)$
At $k = 7$ ,
The binomial likelihood to be assessed is:
$P (X = 7) = (b e g \in {a r r a y} {c} 10 ∖ 7 e n d {a r r a y}) \cdot {(0.55)}^{7} \cdot {(1 - 0.55)}^{10 - 7}$
$= \frac{10!}{7! (10 - 7)!} \cdot {(0.55)}^{7} \cdot {(0.45)}^{3}$
$= 120 \cdot {(0.55)}^{7} \cdot {(0.45)}^{3}$
$\approx 0.1665$
At $k = 8$ ,
The binomial probability that should be assessed is:
$P (X = 8) = (b e g \in {a r r a y} {c} 10 ∖ 8 e n d {a r r a y}) \cdot {(0.55)}^{8} \cdot {(1 - 0.55)}^{10 - 8}$
$= \frac{10!}{8! (10 - 8)!} \cdot {(0.55)}^{8} \cdot {(0.45)}^{2}$
$= 45 \cdot {(0.55)}^{8} \cdot {(0.45)}^{2}$
$\approx 0.0763$
At $k = 9$ ,
The binomial probability to be evaluated as:
$P (X = 9) = (b e g \in {a r r a y} {c} 10 ∖ 9 e n d {a r r a y}) \cdot {(0.55)}^{9} \cdot {(1 - 0.55)}^{10 - 9}$
$= \frac{10!}{9! (10 - 9)!} \cdot {(0.55)}^{9} \cdot {(0.45)}^{1}$
$= 10 \cdot {(0.55)}^{9} \cdot {(0.45)}^{1}$
$\approx 0.0207$

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