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Difficulty: HardTwo-Way Tables and Probability

A fitness center surveyed a random sample of its members about their primary workout time and their membership type. The results of the survey are partially shown in the table below.

Primary Workout TimeBasic MembershipPremium MembershipTotal
Morning6060aa
Afternoonbb20206060
Evening6060140140
Total180180300300

Given that a randomly selected member from this survey has a Basic membership, what is the probability that this member's primary workout time is in the evening?

  1. A
    415\frac{4}{15}
  2. 49\frac{4}{9}Answer
  3. C
    47\frac{4}{7}
  4. D
    29\frac{2}{9}

Answer

49\frac{4}{9}
To find the probability that a randomly selected member's primary workout time is in the evening given that they have a Basic membership, we restrict the denominator to the total number of Basic members, which is 180180. The number of Basic members who work out in the afternoon is 60−20=4060 - 20 = 40. Subtracting the morning (6060) and afternoon (4040) Basic members from the total Basic members (180180) yields 180−60−40=80180 - 60 - 40 = 80 Basic members who work out in the evening. The probability is 80180\frac{80}{180}, which simplifies to 49\frac{4}{9}.

Step-by-Step Solution

1
Find the total number of members with a Premium membership.
The total number of Premium members is 120120.
The grand total of surveyed members is 300300, and the total number of Basic members is 180180. Subtracting the Basic members from the grand total gives the Premium members: 300−180=120300 - 180 = 120.
2
Find the value of bb, which is the number of Basic members whose primary workout time is in the afternoon.
b=40b = 40
The total number of members who work out in the afternoon is 6060, and 2020 of them have a Premium membership. Subtracting the Premium members from the afternoon total gives the Basic members: 60−20=4060 - 20 = 40.
3
Find the number of Basic members whose primary workout time is in the evening.
There are 8080 Basic members who work out in the evening.
The total number of Basic members is 180180. Subtracting the Basic members who work out in the morning (6060) and afternoon (4040) from the total number of Basic members gives the remaining Basic members who work out in the evening: 180−60−40=80180 - 60 - 40 = 80.
4
Calculate the conditional probability that a randomly selected member works out in the evening, given they have a Basic membership.
The probability is 49\frac{4}{9}.
The condition restricts the sample space to the 180180 members with a Basic membership. Of these 180180 members, 8080 work out in the evening. The probability is the ratio of these two values: 80180=49\frac{80}{180} = \frac{4}{9}.

Key Concept

Conditional Probability in Two-Way Tables

Alternative Method

Instead of subtracting from the column total to find the missing Basic Evening value, one could first find all Premium values: the Premium Total is 300−180=120300 - 180 = 120. Then, the Premium Morning value is 120−20−60=40120 - 20 - 60 = 40. This allows calculation of the Morning Total (60+40=10060 + 40 = 100). The Evening Total is given as 140140, and the Premium Evening value is 6060, which directly gives the Basic Evening value as 140−60=80140 - 60 = 80. Finally, the probability is 80180=49\frac{80}{180} = \frac{4}{9}.
Estimated Time:2m 0s
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Topics

Two-Way Tables and Probability

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