Mathematics, 23.10.2020 15:20 loki285

6.45 The length of time for one individual to be served at a cafeteria is a random variable having an exponential distribution with a mean of 4 minutes. What is the probability that a person is served in less than 3 minutes on at least 4 of the next 6 days?

Answers: 2

Show answers

Answers

Answer from: idk5233

Using the exponential and the binomial distribution, it is found that there is a 0.3969 = 39.69% probability that a person is served in less than 3 minutes on at least 4 of the next 6 days.

First, we find the probability that a person is served in less than 3 minutes on a single day, using the exponential distribution.

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

$f(x) = \mu e^{-\mu x}$

In which $\mu = \frac{1}{m}$ is the decay parameter.

The probability that x is lower or equal to a is given by:

$P(X \leq x) = 1 - e^{-\mu x}$

In this problem, mean of 4 minutes, hence $m = 4, \mu = \frac{1}{4} = 0.25$ .

The probability that a person is served in less than 3 minutes on a single day is:

$P(X \leq 3) = 1 - e^{-0.25(3)} = 0.5267$

Now, for the 6 days, we use the binomial distribution.

Binomial probability distribution

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$C_{n,x} = \frac{n!}{x!(n-x)!}$

The parameters are:

x is the number of successes. n is the number of trials. p is the probability of a success on a single trial.

In this problem:

6 days, hence n = 6

On each day, a 0.5276 probability, hence p = 0.5276

The probability is:

$P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6)$

Hence:

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 4) = C_{6,4}.(0.5276)^{4}.(0.4724)^{2} = 0.2594$

$P(X = 5) = C_{6,5}.(0.5276)^{5}.(0.4724)^{1} = 0.1159$

$P(X = 6) = C_{6,6}.(0.5276)^{6}.(0.4724)^{0} = 0.0216$

Then:

$P(X \geq 4) = P(X = 4) + P(X = 5) + P(X = 6) = 0.2594 + 0.1159 + 0.0216 = 0.3969$

0.3969 = 39.69% probability that a person is served in less than 3 minutes on at least 4 of the next 6 days.

A similar problem is given at link

Answer from: taishaun

0.3968

Step-by-step explanation:

For exponential distribution :

P(A < x) = 1 - e^-x/m

Where m = mean of exponential distribution

Mean of distribution = 4

x = 3

P(A < 3) = 1 - e^-3/4

P(A < 3) = 1 - (0.4723665)

P(A < 3) = 0.5276335 =

Hence, p = 0.5276

(1 - p) = 1 - 0.5276 = 0.4724

Binomial distribution :

P(n, x) = nCx * p^x * (1 - p)^(n-x)

Served on atleast 4 of the next 6 days ;

P(4) + P(5) + P(6):

(6C4 * 0.5276^4 * 0.4724^2) + (6C5 * 0.5276^5 * 0.4724^1) + (6C6 * 0.5276^6 * 0.4724^0)

= (15 * 0.5276^4 * 0.4724^2) + (6 * 0.5276^5 * 0.4724^1) + (1 * 0.5276^6 * 0.4724^0)

= 0.3968

Answer from: Quest

profile photo

step-by-step explanation:

Answer from: Quest

No he does not because he can only cover 12 square feet from 15 square feet which he will still need 3 more. this is because if 3×5=15 square feet and 15-12=3 which thats the answer