Make a histogram showing the binomial probability distribution for the following: A waiter at the Green Spot resturant has learned from long experience that the probability that a lone diner will leave a tip is only 0.7. During one lunch hour he serves six people who are dining by themselves. (hint make table showing r= 0,1,2,3,4,5,6 and the p(r) that goes with each one ) What is the area of the bar graph showing r=3?

prob(zero people leave tip) = C(6,0)(.7)^0 (.3)^6 = .000729

prob(1 person leaves tip) = C(6,1)(.7)(.3)^5 = .010206
prob(2 persons leave tip) = C(6,2)(.7)^2 (.3)^4
= .059535
prob(3 persons leave tip) = C(6,3)(.7)^3 (.3)^3
= .18522
prob(4 people leave tip) = C(6,4)(.7)^4 (.3)^2
= .324135
prob(5 people leave tip) = C(6,5) (.7)^5 (.3)
= .302529
prob(6 people leave tip = C(6,6) (.7)^6 (.3)^0
= .117649

note the sum of the above results = 1 , allowing for small round-off error

To make a histogram showing the binomial probability distribution, you need to first calculate the probabilities for each value of r (0, 1, 2, 3, 4, 5, 6).

The probability mass function for a binomial distribution is given by the formula:

P(r) = C(n, r) * p^r * (1-p)^(n-r)

Where:
- n is the number of trials (number of people served, in this case)
- r is the number of successes (number of people leaving a tip, in this case)
- p is the probability of success (probability that a lone diner will leave a tip)

Using this formula, we can calculate the probabilities for each value of r:

r = 0: P(0) = C(6, 0) * 0.7^0 * (1-0.7)^(6-0) = 0.7^6 = 0.1176
r = 1: P(1) = C(6, 1) * 0.7^1 * (1-0.7)^(6-1) = 6 * 0.7^1 * 0.3^5 = 0.3025
r = 2: P(2) = C(6, 2) * 0.7^2 * (1-0.7)^(6-2) = 15 * 0.7^2 * 0.3^4 = 0.3241
r = 3: P(3) = C(6, 3) * 0.7^3 * (1-0.7)^(6-3) = 20 * 0.7^3 * 0.3^3 = 0.1852
r = 4: P(4) = C(6, 4) * 0.7^4 * (1-0.7)^(6-4) = 15 * 0.7^4 * 0.3^2 = 0.0595
r = 5: P(5) = C(6, 5) * 0.7^5 * (1-0.7)^(6-5) = 6 * 0.7^5 * 0.3^1 = 0.0102
r = 6: P(6) = C(6, 6) * 0.7^6 * (1-0.7)^(6-6) = 0.7^6 = 0.1176

Now, we can create a table with the values for r (0, 1, 2, 3, 4, 5, 6) and their corresponding probabilities:

r | p(r)
--------------
0 | 0.1176
1 | 0.3025
2 | 0.3241
3 | 0.1852
4 | 0.0595
5 | 0.0102
6 | 0.1176

To create the histogram, you can use a bar graph with the x-axis representing the values of r and the y-axis representing the probabilities. The height of each bar represents the probability for the corresponding value of r.

To find the area of the bar graph showing r = 3, you need to calculate the area under the bar representing r = 3. Since this is a discrete distribution (binomial), the area can be calculated as the product of the height (probability) and the width of the bar.

From the table, we know that p(r=3) = 0.1852. The width of each bar is typically taken as 1. Therefore, the area of the bar graph showing r = 3 is 0.1852.