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Sagot :
The required probabilities are calculated by using the binomial distribution formula as below:
a. The probability that 5 will still be with the company after 1 year is P(x = 5) = 0.2541
b. The probability that 5 or more still be with the company after 1 year is 0.8057.
What is the binomial distribution formula?
The binomial distribution formula for finding the required probability of a random variable is
[tex]P(X = r) = _nC_rp^rq^{n-r}[/tex]
Where
[tex]_nC_r = \frac{n!}{r!(n-r)!}[/tex]
p = probability of success
q = 1 - p = probability of failure
n = total number of trials
r = number of trials chosen
Calculation:
It is given that,
n = 8 and p = 0.7
Then, q = 1 - 0.7 = 0.3
a. The probability that 5 will still be with the company after 1 year:
P(X = 5) = [tex]_8C_5[/tex] (0.7)⁵(0.3)⁸⁻⁵
⇒ [tex]\frac{8!}{5!(8-5)!}[/tex] (0.7)⁵(0.3)³
⇒ 0.2541
Therefore, the probability P(x = 5) is 0.254.
b. The probability that 5 or more still be with the company after 1 year:
More than 5 means 6, 7, and 8
So,
P(X = 6) = [tex]_8C_6[/tex] (0.7)⁶(0.3)⁸⁻⁶
⇒ [tex]\frac{8!}{6!(8-6)!}[/tex] (0.7)⁶(0.3)²
⇒ 0.2964
P(X = 7) = [tex]_8C_7[/tex] (0.7)⁷(0.3)⁸⁻⁷
⇒ [tex]\frac{8!}{7!(8-7)!}[/tex] (0.7)⁷(0.3)¹
⇒ 0.1976
P(X = 8) = [tex]_8C_8[/tex] (0.7)⁸(0.3)⁸⁻⁸
⇒ [tex]\frac{8!}{8!(8-8)!}[/tex] (0.7)⁸(0.3)⁰
⇒ 0.0576
Thus, the probability that 5 or more still be with the company after 1 year
= P(5) + P(6) + P(7) + P(8)
= 0.2541 + 0.2964 + 0.1976 + 0.0576
= 0.8057.
Therefore, the required probabilities are 0.2541 and 0.8057.
Disclaimer: The question given in the portal is incomplete. Here is the complete question.
Question: If the probability of a new employee in a fast-food chain still being with the company at the end of the year is 0. 7, what is the probability that out of 8 new hired people:
a. 5 will still be with the company after 1 year?
b. 5 or more still be with the company after 1 year?
Learn more about binomial distribution here:
https://brainly.com/question/15246027
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