1) If the probability of a machine producing a defective part is 0.05, what is the probability of finding exactly 5 defective parts from a sample of 100? (Assume that the process follows a binomial distribution and round answer to four places).
2) If 20% of the people in a community use the emergency room at a hospital in one year, find the following probability for a sample of 10 people.
a) At most three used the emergency room
b) Exactly three used the emergency room
c) At least five used the emergency room
3) It was found that 60% of Ghanaian victims of COVID 19 are senior citizens. If 10 victims are randomly selected, find the probability that exactly three are senior citizens.
Answers
Given : probability of a machine producing a defective part is 0.05,
60% of Ghanaian victims of COVID 19 are senior citizens
To find : what is the probability of finding exactly 5 defective parts from a sample of 100
If 10 victims are randomly selected, find the probability that exactly three are senior citizens.
Solution:
probability of a machine producing a defective part is 0.05,
p = 0.05
q = 1 - p = 0.95
n = 100
x = 5
P(x) = ⁿCₓpˣqⁿ⁻ˣ
=> P(5) = ¹⁰⁰C₅ (0.05)⁵(0.95)⁹⁵
= 0.18
0.18 is the probability that exactly 5 defective parts from a sample of 100
another example same as above :
20% of the people in a community use the emergency room at a hospital in one year
=> p = 0.2 q = 0.8 , n = 10
a) At most three used the emergency room
P(0) + P(1) + P(2) + P(3)
b) Exactly three used the emergency room find P(3)
c) At least five used the emergency room
P(5) + P(6) + P(7) + P(8) + P(9) + P(10)
3) It was found that 60% of Ghanaian victims of COVID 19 are senior citizens.
p = 0.6 q = 0.4 n = 10
P(3) = ¹⁰C₃(0.6)³(0.4)⁷ = 0.042
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