Question

According to a recent by the CDC, 37% of Americans said that they'd eaten fast food within the past 24 hrs with 7.8% of Americans being considered High income AND have eaten fast food within the past 24HRS. It's also know that 21.1% of Americans are High Income, 49.9% are Middle Income, and the rest of individuals are Low Income. Finally, 18.0% of Americans are considered Middle Income AND have eaten fast food within the past 24 HRS.
A) calculated the probability that someone has eaten fast food in the last 24 HRS given that they're Middle Income.
B) calculated the probability that someome has eaten fast food or is low income
C) Are the events of eating fast food and income independent? If not, please briefly explain how the events are related

Answer 1

Here,

P[ eaten fast food within the past 24 hrs] = P[E] = 0.37

P[High Income AND eaten fast food] = P[H and E] = 0.078

P[  Middle Income AND eaten fast food ] = P[M and E] = 0.18

P[High Income]=P[H] =0.211

P[Middle Income]=P[M] =0.499

P[Low Income]= P[L] = 1- 0.211 - 0.499 = 0.29

A)

Note: In case of dependent events , the probability that both events occur simultaneously is:

P(A and B)=P(A)⋅P(B | A) (Conditional probability)

So, probability that someone has eaten fast food in the last 24 HRS given that they're Middle Income.

P[E/H] = P[H and E] / P[H] = 0.078 / 0.211 = 0.3697

B) The probability that someome has eaten fast food or is low income

P[E or L] = P[E] + P[L] - P[E and L] = P[E] + P[L] - P[E].P[L]

= 0.37 + 0.29 + 0.37*0.29

= 0.66 - 0.1073 = 0.5527

C) YES

Two events are said to be independent, when the occurrence of one event cannot control the occurrence of other. For independent events P(A and B) = P(A) P(B)

Here the  the events of eating fast food and income are clearly independent.