Answer to Question #166418 in Statistics and Probability for Sky Escudero

Question

1. The checking accounts of Sun Bank are categorized by the age of account and the account balance. Auditor will select accounts at random from the following 1000 accounts

a)    What is the probability that an account is less than 2 years old?

b)    What is the probability that an account has balance of $1000 or more?

c)    What is the probability that the two accounts will both have a balance of $1000 or more?

d)    What is the probability that an account has a balance of $500-$999 given that its age is 2 years or more?

e)    What is the probability that an account is less than 2 years old and has a balance of $1000 or more?

f)      What is the probability that an account is at least 2 years old given that the balance is $500-$999?

Answer 1

Answer:

Given data:

The total numbers of accounts are n=1000.

The numbers of accounts that are less than 2 years and that have a balance greater than $1000 are a=90.

The numbers of accounts that are more than 2 years and have a balance greater than $1000 are b=200.

The numbers of accounts that are more than 2 years and have a balance between $500-$999 are c=275.

The numbers of accounts that are more than 2 years and have a balance between $0-$499 are d=75.

The numbers of accounts that are more than 2 years and have a balance greater than $1000 are e=200.

a)

Probability = $120 + 240 + 90 \over 1000$

Probability = $450 \over 1000$

Probability = 0.45 or 45%

b)

Probability = $200 + 90 \over 1000$

Probability = $290 \over 1000$

Probability = 0.29 or 29%

c)

The expression for the probability that two accounts are selected that have a balance greater than $1000 is,

P(A)=( $A+B \over n$) ( $A+B \over n$)

Substitute the given values in the above expression.

P(A)=( $90+200 \over 1000$) ( $90+200 \over 1000$)

P(A)= (0.29) (0.29)

P(A)= 0.0841

Thus, the probability that two accounts are selected that have a balance greater than $1000 is 0.0841.

d)

P(B) = $c \over d+c+e$

Substitute the given values in the above expression.

P(B) = $275 \over 75+275+200$

P(B) = $275 \over 550$

P(B) = 0.5

Thus, the probability that the account balance is between $500-$999 and given that it is 2 years old is 0.5.

e)

The probability that an account is less than 2 years old and has a balance of $1000 or more is,

P(less than 2 years AND $1000 or More) = ${n(less than 2 years AND 1000 or More)} \over {n(total)}$

P(less than 2 years AND $1000 or More) = $90 \over 1000$

P(less than 2 years AND $1000 or More) = 0.09

Thus, the probability that an account is less than 2 years old and has a balance of $1000 or more is 0.09.

f)

The probability that an account is at least 2 years old given that the balance is $500-$999 is,

P(at least 2 years GIVEN $500−$999) = $P (at least 2 years AND 500−999) \over {P(500−999)}$

P(at least 2 years GIVEN $500−$999) = ${275 \over 1000} \over {515 \over 1000}$

P(at least 2 years GIVEN $500−$999) = $275 \over 515$

P(at least 2 years GIVEN $500−$999) = 0.5340

Thus, the probability that an account is at least 2 years old given that the balance is $500-$999 is 0.5340.