Answer :
Certainly! Let's go through each part of the problem step-by-step.
(i) Probability that the person is under 21 years:
To find the probability that a randomly selected person is under 21 years old, we sum up all the probabilities associated with the "Under 21" category across all the visit categories (0 visits, 1 visit, and 2+ visits).
[tex]\[ P(\text{Under 21}) = 0.04 + 0.06 + 0.02 = 0.12 \][/tex]
So, the probability that the person is under 21 years is 0.12.
---
(ii) Probability that the person visited at least twice in a month:
To find the probability that a person visited the supermarket at least twice in a month, we sum up the probabilities of "2+ visits" across all the age categories.
[tex]\[ P(\text{at least 2 visits}) = 0.02 + 0.01 + 0.02 + 0.01 = 0.06 \][/tex]
So, the probability that a person visited at least twice in a month is 0.06.
---
(iii) Probability that the person has visited at least twice in a month, given he is under 21 years:
To find this conditional probability, we will use the formula for conditional probability:
[tex]\[ P(\text{2+ visits | Under 21}) = \frac{P(\text{Under 21 and 2+ visits})}{P(\text{Under 21})} \][/tex]
We know from the table that:
[tex]\[ P(\text{Under 21 and 2+ visits}) = 0.02 \][/tex]
And from part (i), we have:
[tex]\[ P(\text{Under 21}) = 0.12 \][/tex]
Therefore,
[tex]\[ P(\text{2+ visits | Under 21}) = \frac{0.02}{0.12} = 0.16666666666666669 \][/tex]
So, the probability that a person has visited at least twice in a month, given they are under 21 years, is 0.16666666666666669.
---
(iv) Probability that the person has visited once:
To find the probability that a person has visited the supermarket once, we sum up the probabilities of "1 visit" across all the age categories.
[tex]\[ P(\text{1 visit}) = 0.06 + 0.16 + 0.10 + 0.03 = 0.35 \][/tex]
So, the probability that a person has visited once is 0.35.
---
(v) Probability that the person is between 41-60 years given they visited once:
To find this conditional probability, we will use the formula for conditional probability:
[tex]\[ P(\text{41-60 | 1 visit}) = \frac{P(\text{1 visit and 41-60})}{P(\text{1 visit})} \][/tex]
We know from the table that:
[tex]\[ P(\text{1 visit and 41-60}) = 0.10 \][/tex]
And from part (iv), we have:
[tex]\[ P(\text{1 visit}) = 0.35 \][/tex]
Therefore,
[tex]\[ P(\text{41-60 | 1 visit}) = \frac{0.10}{0.35} = 0.28571428571428575 \][/tex]
So, the probability that the person is between 41-60 years, given they visited once, is 0.28571428571428575.
---
(vi) Probability that the person has not visited the supermarket in the last month:
To find the probability that a person has not visited the supermarket in the last month, we sum up the probabilities of "0 visits" across all the age categories.
[tex]\[ P(\text{0 visits}) = 0.04 + 0.25 + 0.23 + 0.08 = 0.6 \][/tex]
So, the probability that a person has not visited the supermarket in the last month is 0.6.
---
(vii) Probability that the person has not visited the supermarket, given they are between 21-40 years:
To find this conditional probability, we will use the formula for conditional probability:
[tex]\[ P(\text{0 visits | 21-40}) = \frac{P(\text{21-40 and 0 visits})}{P(\text{21-40})} \][/tex]
We know from the table that:
[tex]\[ P(\text{21-40 and 0 visits}) = 0.25 \][/tex]
To find [tex]\(P(\text{21-40})\)[/tex], we sum up all the probabilities for the "21-40" age category:
[tex]\[ P(\text{21-40}) = 0.25 + 0.16 + 0.01 = 0.42 \][/tex]
Therefore,
[tex]\[ P(\text{0 visits | 21-40}) = \frac{0.25}{0.42} = 0.5952380952380952 \][/tex]
So, the probability that a person has not visited, given they are between 21-40 years, is 0.5952380952380952.
---
To summarize:
(i) 0.12
(ii) 0.06
(iii) 0.16666666666666669
(iv) 0.35
(v) 0.28571428571428575
(vi) 0.6
(vii) 0.5952380952380952
(i) Probability that the person is under 21 years:
To find the probability that a randomly selected person is under 21 years old, we sum up all the probabilities associated with the "Under 21" category across all the visit categories (0 visits, 1 visit, and 2+ visits).
[tex]\[ P(\text{Under 21}) = 0.04 + 0.06 + 0.02 = 0.12 \][/tex]
So, the probability that the person is under 21 years is 0.12.
---
(ii) Probability that the person visited at least twice in a month:
To find the probability that a person visited the supermarket at least twice in a month, we sum up the probabilities of "2+ visits" across all the age categories.
[tex]\[ P(\text{at least 2 visits}) = 0.02 + 0.01 + 0.02 + 0.01 = 0.06 \][/tex]
So, the probability that a person visited at least twice in a month is 0.06.
---
(iii) Probability that the person has visited at least twice in a month, given he is under 21 years:
To find this conditional probability, we will use the formula for conditional probability:
[tex]\[ P(\text{2+ visits | Under 21}) = \frac{P(\text{Under 21 and 2+ visits})}{P(\text{Under 21})} \][/tex]
We know from the table that:
[tex]\[ P(\text{Under 21 and 2+ visits}) = 0.02 \][/tex]
And from part (i), we have:
[tex]\[ P(\text{Under 21}) = 0.12 \][/tex]
Therefore,
[tex]\[ P(\text{2+ visits | Under 21}) = \frac{0.02}{0.12} = 0.16666666666666669 \][/tex]
So, the probability that a person has visited at least twice in a month, given they are under 21 years, is 0.16666666666666669.
---
(iv) Probability that the person has visited once:
To find the probability that a person has visited the supermarket once, we sum up the probabilities of "1 visit" across all the age categories.
[tex]\[ P(\text{1 visit}) = 0.06 + 0.16 + 0.10 + 0.03 = 0.35 \][/tex]
So, the probability that a person has visited once is 0.35.
---
(v) Probability that the person is between 41-60 years given they visited once:
To find this conditional probability, we will use the formula for conditional probability:
[tex]\[ P(\text{41-60 | 1 visit}) = \frac{P(\text{1 visit and 41-60})}{P(\text{1 visit})} \][/tex]
We know from the table that:
[tex]\[ P(\text{1 visit and 41-60}) = 0.10 \][/tex]
And from part (iv), we have:
[tex]\[ P(\text{1 visit}) = 0.35 \][/tex]
Therefore,
[tex]\[ P(\text{41-60 | 1 visit}) = \frac{0.10}{0.35} = 0.28571428571428575 \][/tex]
So, the probability that the person is between 41-60 years, given they visited once, is 0.28571428571428575.
---
(vi) Probability that the person has not visited the supermarket in the last month:
To find the probability that a person has not visited the supermarket in the last month, we sum up the probabilities of "0 visits" across all the age categories.
[tex]\[ P(\text{0 visits}) = 0.04 + 0.25 + 0.23 + 0.08 = 0.6 \][/tex]
So, the probability that a person has not visited the supermarket in the last month is 0.6.
---
(vii) Probability that the person has not visited the supermarket, given they are between 21-40 years:
To find this conditional probability, we will use the formula for conditional probability:
[tex]\[ P(\text{0 visits | 21-40}) = \frac{P(\text{21-40 and 0 visits})}{P(\text{21-40})} \][/tex]
We know from the table that:
[tex]\[ P(\text{21-40 and 0 visits}) = 0.25 \][/tex]
To find [tex]\(P(\text{21-40})\)[/tex], we sum up all the probabilities for the "21-40" age category:
[tex]\[ P(\text{21-40}) = 0.25 + 0.16 + 0.01 = 0.42 \][/tex]
Therefore,
[tex]\[ P(\text{0 visits | 21-40}) = \frac{0.25}{0.42} = 0.5952380952380952 \][/tex]
So, the probability that a person has not visited, given they are between 21-40 years, is 0.5952380952380952.
---
To summarize:
(i) 0.12
(ii) 0.06
(iii) 0.16666666666666669
(iv) 0.35
(v) 0.28571428571428575
(vi) 0.6
(vii) 0.5952380952380952
