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A health survey was conducted among 400 adults to examine their dietary habits. The survey revealed that 250 adults reported consuming fruits regularly, while 150 adults reported consuming vegetables regularly. Among the 300 adults who exercised daily, 180 also reported consuming fruits regularly.

a) Display the information in a two-way classification table.

b) A person is randomly selected from the 400 surveyed. What is the probability that:

i) The person consumes fruits regularly but does not exercise daily?

ii) The person exercises daily and consumes fruits regularly?

c) What is the probability that a randomly selected person consumes fruits regularly given that they exercise daily?

d) If a person does not consume fruits regularly, what is the probability that they exercise daily?

Answer :

To tackle the health survey problem, we need to use basic principles of probability and set notation. Let's go through each part step-by-step.

Given Data

  1. Total adults surveyed: 400
  2. Adults consuming fruits regularly: 250
  3. Adults consuming vegetables regularly: 150
  4. Adults exercising daily: 300
  5. Adults exercising daily and consuming fruits regularly: 180

a) Two-way Classification Table

Let's organize the data using a two-way table involving exercising and consuming fruits:

Fruits Regularly (Yes)Fruits Regularly (No)TotalExercises Daily (Yes)180120300Exercises Daily (No)7030100Total250150400

Note: The figures for "Exercises Daily (No)" row and "Fruits Regularly (No)" column are calculated based on the totals.

b) Probability Calculations

i) The probability that the person consumes fruits regularly but does not exercise daily

  • There are 70 people who consume fruits regularly but do not exercise daily.
  • Probability = [tex]\frac{70}{400} = 0.175[/tex]

ii) The probability that the person exercises daily and consumes fruits regularly

  • There are 180 people who exercise daily and consume fruits regularly.
  • Probability = [tex]\frac{180}{400} = 0.45[/tex]

c) Probability of Consuming Fruits Regularly given Exercising Daily

  • We want the probability that a person consumes fruits regularly given that they exercise daily.
  • Formula: [tex]P(F \mid E) = \frac{P(F \cap E)}{P(E)}[/tex]
  • [tex]P(F \cap E) = \frac{180}{400}[/tex], [tex]P(E) = \frac{300}{400}[/tex]
  • [tex]P(F \mid E) = \frac{180/400}{300/400} = \frac{180}{300} = 0.6[/tex]

d) If a person does not consume fruits regularly, what is the probability they exercise daily

  • We use the complementary probability.
  • Total people who do not consume fruits regularly = 150.
  • Out of these, 120 exercise daily.
  • Probability = [tex]\frac{120}{150} = 0.8[/tex]

Each part uses conditional probability and basic counting to find the required probabilities. This systematic approach helps solve problems involving two-way tables and conditional probability.

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