Answer:
a) P(X = 4) = 0.1655
b) P(X < 3) = 0.4752
c) P(Y < 5) = 0.0273
Step-by-step explanation:
Hello!
The study variable is
X: Number of people that do not cover their mouths when they sneeze, in a sample of 10.
This is a discrete variable with two possible outcomes "people do not cover their mouth when they sneeze" (success) "people cover their mouth when they sneeze" (failure). First, let's check if it follows the Binomial criteria:
The number of observation of the trial is fixed (n=10)
Each observation in the trial is independent, this means that none of the trials will have an effect on the probability of the next trial (Whether one person covers or not its mouth when he sneezes doesn't modify the probability of the next one not covering its mouth when sneezing)
The probability of success in the same from one trial to another (ρ= 0.267)
So X≈ Bi (n;ρ)
since the study variable has binomial distribution, I'll use the tables of the binomial distribution to calculate the asked probabilities. (Remember, these tables give values of cumulative probabilities; P(X≤r))
a) What is the probability that among 10 randomly observed individuals exactly 4 do not cover their mouth when sneezing?
P(X = 4) = P(X ≤ 4) - P(X ≤ 3) = 0.9004 - 0.7349 = 0.1655
b) What is the probability that among 10 randomly observed individuals fewer than 3 do not cover their mouth when sneezing?
P(X < 3) = P(X ≤ 2) = 0.4752
c) Would you be surprised if, after observing 10 individuals, fewer than half covered their mouths when sneezing?
If fewer than half covered their mouths, then more than half didn't cover their mouths. (I'll call Y: the number of people that covered their mouths while sneezing) Symbolically:
P(Y < 5) ⇒ P(X > 5)
P(X > 5) = 1 - P(X ≤ 5) = 1 - 0.9727 = 0.0273
I hope you have a SUPER day!
