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Answer:
a) the probability that a randomly selected person fails to meet the activity guidelines or eats a poor diet is 0.83
b) the probability that a randomly selected person is leading a healthy lifestyle is 0.17
c) the probability that they all 4 people eat a poor diet assuming independence is 0.05308
d) the probability that the first one meets the Physical Activity Guidelines and the last two do not is 0.1364
Step-by-step explanation:
Given the data in the question;
Let E represent the events
E1 = P( person fails to meet CDC Physical Activity Guidelines ) = 0.77
E2 = P( person eats a poor diet ) = 0.48
E3 = P(E1 ∩ E2) = P( both ) = 42
a) What is the probability that a randomly selected person fails to meet the activity guidelines or eats a poor diet?
the probability that a person eats a poor diet or fails to meet the activity guidelines will be;
P( E1 ∪ E2 ) = P(E1) + P(E2) - P(E1 ∩ E2)
so we substitute
P( E1 ∪ E2 ) = 0.77 + 0.48 - 0.42 = 0.83
Therefore, the probability that a randomly selected person fails to meet the activity guidelines or eats a poor diet is 0.83
b) Find the probability that a randomly selected person is leading a healthy lifestyle. That is, they meet the Physical Activity Guidelines and eat a diet that is not poor?
In the light of De-Morgan’s Laws
P( ∩
) = 1 - P( E1 ∪ E2 )
so
P( ∩
) = 1 - 0.83 = 0.17
Therefore, the probability that a randomly selected person is leading a healthy lifestyle is 0.17
c) If you randomly select 4 people, what is the probability that they all eat a poor diet assuming independence?
if we select 4 people;
E2 = P( person eats a poor diet ) = 0.48
so
P = ( 0.48 )⁴ = 0.05308
Therefore, the probability that they all 4 people eat a poor diet assuming independence is 0.05308
d) If you randomly select 3 people, what is the probability that the first one meets the Physical Activity Guidelines and the last two do not?
we select 3 people;
P( person fails to meet CDC Physical Activity Guidelines ) = 0.77
P( person meet CDC Physical Activity Guidelines ) = 1 - 0.77 = 0.23
so
P = 0.23 × ( 0.77)²
P = 0.1364
Therefore, the probability that the first one meets the Physical Activity Guidelines and the last two do not is 0.1364