Question

A food truck near the beach sells the best rock lobster tacos. 75% of the time, people will choose the rock lobster tacos vs something else. The cooks need help calculating the chances of the following situations. First, find the mean and standard deviation of this distribution given there are 5 customers waiting in line. Use 4 decimals. 

What is the chance of selling none of the customers the rock lobster tacos given there are 5 customers in line? Use 5 decimals. 

Now, what is the probability of selling the rock lobster tacos to at least one customer of the 5 in line? Use hand calculations and use 5 decimals for your answer. 

Using the appropriate StatCrunch calculator, fill it out to show the probability of selling the tacos to 3 to 8 customers out of 12 customers in line. Include the graph in your answer. Use 4 decimals.

Answer 1

P(customer choses rock lobster ) = 0.75 = p = probability of success

P(customer does not choses rock lobster ) = 1 - 0.75 =0.25 = 1 - p = probability of failure

As we will use binomial distribution , we have mentioned p and q in above statements only

P(X=x) in binomial ditribution = nCx * p^x * (1-p)^(n-x)

mean = sum of (x * P(X=x)

variance = sum of (x² * P(X=x) - {sum of (x * P(X=x)}²

Standard deviation = variance ^0.5

	x	P(X=x)	x*P(X=x)	x²*P(X=x)
	0	0.000977	0	0
	1	0.014648	0.014648	0.014648
	2	0.087891	0.175781	0.351563
	3	0.263672	0.791016	2.373047
	4	0.395508	1.582031	6.328125
	5	0.237305	1.186523	5.932617
	sum	1	3.75	15
	mean	3.75
	variance	0.9375
	SD	0.968246

P(Selling none of the customers the rock lobster tacos given there are 5 customers in line) = 0.000977

P(what is the probability of selling the rock lobster tacos to at least one customer of the 5 in line) = 1- P(Selling none of the customers the rock lobster tacos given there are 5 customers in line)

P(what is the probability of selling the rock lobster tacos to at least one customer of the 5 in line) = 1-0.000977 = 0.999023

Now , taking 12 customers , we get

x	P(X=x)
0	5.96046E-08
1	2.14577E-06
2	3.54052E-05
3	0.000354052
4	0.002389848
5	0.011471272
6	0.04014945
7	0.103241444
8	0.193577707
9	0.258103609
10	0.232293248
11	0.126705408
12	0.031676352
sum	1

P( selling the tacos to 3 to 8 customers out of 12 customers in line) = 0.000354 + 0.0002389 + 0.011471272 + 0.04014945 + 0.10324144 + 0.1935777 = 0.351184