Chelsea and her friends ate 12 of 16 apples she purchased. What percent of apples did they eat?
2 answers:
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<h3>Answer:</h3>
(x, y) = (7, -5)
<h3>Step-by-step explanation:</h3>It generally works well to follow directions.
The matrix of coefficients is ...
Its inverse is the transpose of the cofactor matrix, divided by the determinant. That is ...
So the solution is the product of this and the vector of constants [-6, -50]. That product is ...
... x = (3·(-6) +(-4)(-50))/26 = 7
... y = (5·(-6) +2·(-50))/26 = -5
The solution using inverse matrices is ...
... (x, y) = (7, -5)
An interesting twist to a binomial distribution problem.
Given:
p=55%=0.55 for probability of success in solicitation
x=4=number of successful solicitations
n=number of calls to be made
P(x,n,p)>=89.9%=0.899 (from context, it is >= and not =, which is almost impossible)
From context of question, all calls are assumed independent, with constant probability of success, so binomial distribution is applicable.
The number of successes, x, is then given by
where
p=probability of success
n=number of trials
x=number of successes
Here we need n such that
P(x,n,p)>=0.899
given
x>=4, p=0.55, which means we need to find
Method 1: if a cumulative binomial distribution table is available, we can look up n=9,10,11 and find
P(x>=4,9,0.55)=0.834
P(x>=4,10,0.55)=0.898
P(x>=4,11,0.55)=0.939
So she must make (at least) 11 calls to make sure the probability of meeting her quota is 89.9% or more.
Method 2: using technology.
Similar to method 1, we can look up the probabilities directly, for n=9,10,11
P(x>=4,9,0.55)=0.834178
P(x>=4,10,0.55)=0.8980051
P(x>=4,11,0.55)=0.9390368
Method 3: using simple calculator
Here we need to calculate the probabilities for each value of n=10,11 and sum the probabilities of FAILURE S=P(0,n,0.55)+P(1,n,0.55)+P(2,n,0.55)+P(3,n,0.55)
so that the probability of success is 1-S.
For n=10,
P(0,10,0.55)=0.000341
P(1,10,0.55)=0.004162
P(2,10,0.55)=0.022890
P(3,10,0.55)=0.074603
So that
S=0.000341+0.004162+0.022890+0.074603
=0.101995
and Probability of getting 4 successes (or more)
=1-S
=0.898005, missing target by 0.1%
So she will have to make 11 phone calls, bring up the probability to 93.9%. The work is similar to that of n=10.
Given:
p=55%=0.55 for probability of success in solicitation
x=4=number of successful solicitations
n=number of calls to be made
P(x,n,p)>=89.9%=0.899 (from context, it is >= and not =, which is almost impossible)
From context of question, all calls are assumed independent, with constant probability of success, so binomial distribution is applicable.
The number of successes, x, is then given by
p=probability of success
n=number of trials
x=number of successes
Here we need n such that
P(x,n,p)>=0.899
given
x>=4, p=0.55, which means we need to find
Method 1: if a cumulative binomial distribution table is available, we can look up n=9,10,11 and find
P(x>=4,9,0.55)=0.834
P(x>=4,10,0.55)=0.898
P(x>=4,11,0.55)=0.939
So she must make (at least) 11 calls to make sure the probability of meeting her quota is 89.9% or more.
Method 2: using technology.
Similar to method 1, we can look up the probabilities directly, for n=9,10,11
P(x>=4,9,0.55)=0.834178
P(x>=4,10,0.55)=0.8980051
P(x>=4,11,0.55)=0.9390368
Method 3: using simple calculator
Here we need to calculate the probabilities for each value of n=10,11 and sum the probabilities of FAILURE S=P(0,n,0.55)+P(1,n,0.55)+P(2,n,0.55)+P(3,n,0.55)
so that the probability of success is 1-S.
For n=10,
P(0,10,0.55)=0.000341
P(1,10,0.55)=0.004162
P(2,10,0.55)=0.022890
P(3,10,0.55)=0.074603
So that
S=0.000341+0.004162+0.022890+0.074603
=0.101995
and Probability of getting 4 successes (or more)
=1-S
=0.898005, missing target by 0.1%
So she will have to make 11 phone calls, bring up the probability to 93.9%. The work is similar to that of n=10.