Question

Twenty-five percent of the customers entering a grocery store between 5 P.M. and 7 P.M. use an express checkout. Consider five randomly selected customers, and let x denote the number among the five who use the express checkout.(a) What is p(1), that is, P(x = 1)? (Round the answer to five decimal places.)p(1) =__________(b) What is P(x 1)? (Round the answer to five decimal places.)P(x 1) =__________(c) What is P(2 x)? (Round the answer to five decimal places. Hint: Make use of your computation in Part (b).)P(2 x) =________(d) What is P(x 1)? (Round the answer to five decimal places.)P(x 1) =__________

Answer 1

Answer:

a) $P(X=1)=(5C1)(0.25)^1 (1-0.25)^{5-1}=0.39551$

b)  $P(X \leq 1) = P(X=0) +P(X=1)=0.23730+0.39551 =0.63281$

c) $P(X \geq 2) = 1-P(X$

And for this case we can use the result from part b

$P(X \geq 2) = 1-P(X$

d) $P(X \neq 1)$ since the random variable just takes values between 0 and 5 we can use the complement rule like this:

$P(X \neq 1) = 1-P(X=1)= 1-0.39551=0.60449$

Step-by-step explanation:

Previous concepts

A Bernoulli trial is "a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted". And this experiment is a particular case of the binomial experiment.

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:  

$P(X)=(nCx)(p)^x (1-p)^{n-x}$  

Where (nCx) means combinatory and it's given by this formula:

$nCx=\frac{n!}{(n-x)! x!}$  

The complement rule is a theorem that provides a connection between the probability of an event and the probability of the complement of the event. Lat A the event of interest and A' the complement. The rule is defined by: $P(A)+P(A') =1$

Part a

For this case our random variable X who represent the "number among the five who use the express checkout." follows $X \sim Bin (n=5, p=0.25)$

And we can find P(X=1) replacing on the mass function like this:

$P(X=1)=(5C1)(0.25)^1 (1-0.25)^{5-1}=0.39551$

Part b

For this case assuming that we want to find this probability $P(X \leq 1)$ we can do this:

$P(X \leq 1) = P(X=0) +P(X=1)$

$P(X=0)=(5C0)(0.25)^0 (1-0.25)^{5-0}=0.23730$

$P(X=1)=(5C1)(0.25)^1 (1-0.25)^{5-1}=0.39551$

$P(X \leq 1) = P(X=0) +P(X=1)=0.23730+0.39551 =0.63281$

Part c

We can find $P(2 \leq X)$ replacing on the mass function like this, using the complement rule:

$P(X \geq 2) = 1-P(X$

And for this case we can use the result from part b

$P(X \geq 2) = 1-P(X$

Part d

Assuming that we want to find this probability:

$P(X \neq 1)$ since the random variable just takes values between 0 and 5 we can use the complement rule like this:

$P(X \neq 1) = 1-P(X=1)= 1-0.39551=0.60449$