A fitness center surveyed a random sample of its members about their primary workout time … — SAT

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 Time	Basic Membership	Premium Membership	Total
Morning	$60$	$a$
Afternoon	$b$	$20$	$60$
Evening	$60$	$140$
Total	$180$	$300$

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?

A
$\frac{4}{15}$
$\frac{4}{9}$ Cevap
C
$\frac{4}{7}$
D
$\frac{2}{9}$

Cevap

\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

180

. The number of Basic members who work out in the afternoon is

60 - 20 = 40

. Subtracting the morning (

60

) and afternoon (

40

) Basic members from the total Basic members (

180

) yields

180 - 60 - 40 = 80

Basic members who work out in the evening. The probability is

\frac{80}{180}

, which simplifies to

\frac{4}{9}

Adım Adım Çözüm

Find the total number of members with a Premium membership.

The total number of Premium members is

120

The grand total of surveyed members is

300

, and the total number of Basic members is

180

. Subtracting the Basic members from the grand total gives the Premium members:

300 - 180 = 120

Find the value of

b

, which is the number of Basic members whose primary workout time is in the afternoon.

b = 40

The total number of members who work out in the afternoon is

60

, and

20

of them have a Premium membership. Subtracting the Premium members from the afternoon total gives the Basic members:

60 - 20 = 40

Find the number of Basic members whose primary workout time is in the evening.

There are

80

Basic members who work out in the evening.

The total number of Basic members is

180

. Subtracting the Basic members who work out in the morning (

60

) and afternoon (

40

) from the total number of Basic members gives the remaining Basic members who work out in the evening:

180 - 60 - 40 = 80

Calculate the conditional probability that a randomly selected member works out in the evening, given they have a Basic membership.

The probability is

\frac{4}{9}

The condition restricts the sample space to the

180

members with a Basic membership. Of these

180

members,

80

work out in the evening. The probability is the ratio of these two values:

\frac{80}{180} = \frac{4}{9}

Anahtar Kavram

Conditional Probability in Two-Way Tables

Alternatif Yöntem

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 = 120

. Then, the Premium Morning value is

120 - 20 - 60 = 40

. This allows calculation of the Morning Total (

60 + 40 = 100

). The Evening Total is given as

140

, and the Premium Evening value is

60

, which directly gives the Basic Evening value as

140 - 60 = 80

. Finally, the probability is

\frac{80}{180} = \frac{4}{9}

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Cevap

Adım Adım Çözüm

Anahtar Kavram

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Alternatif Yöntem

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