A restaurant needs a staff of 3 waiters and 2 chefs to be properly staffed. The joint probability model for the number of waiters (X) and chefs (Y) that show up on any given day is given below. X
0 1 2 3
Y 0 K 0.02 0.01 0.03 1
1 0.01 0.03 0.05 0.04 2
2 0.02 0.01 0.04 0.69
A restaurant needs a staff of 3 waiters and 2 chef
a) What must the value of k be for this to be a valid probability model?
b) What is the probability that at least one waiter and at least one chef show up on any given day?
c) What is the probability that more chefs show up than waiters on any given day?
d) What is the probability that more than three total staff (waiters and chefs) will show up on any given day?

a) The value of k that mass this a valid pribability model = 0.04

b) The probability that at least one waiter and at least one chef show up on any given day = 0.86

c) The probability that more chefs show up than waiters on any given day = 0.04

d) The probability that more than three total staff (waiters and chefs) will show up on any given day = 0.77

Step-by-step explanation:

The table of probabilities represents the combined probabilities of whatever number of waiters and chef.

The joint probability model for the number of waiters (X) and chefs (Y), including the 'total' row amd column, showing the sum of the joint probabilities along a row or column

Note That, for this to be a valid probability model, all of the joint probabilities must sum up to give 1.00

Y/X | 0 | 1 | 2 | 3 | Total

0 | K | 0.02 | 0.01 | 0.03 | (k+0.07)

1 | 0.01 | 0.03 | 0.05 | 0.04 | 0.13

2 | 0.02 | 0.01 | 0.04 | 0.69 | 0.76

T | (k+0.03) | 0.07 | 0.10 | 0.76 | 1.00

Note; (Y/X) just shows that Y is for the vertical and X takes the horizontal. And T = total along the vertical.

a) Using either the total along the vertical or horizontal, it should end up at the same answer of 1.00 if this is truly a valid probability model.

(k+0.07) + 0.13 + 0.76 = 1.00

or

(k+0.03) + 0.07 + 0.10 + 0.76 = 1.00

Either ways,

k = 1.00 - 0.96 = 0.04

The table then becomes

Y/X | 0 | 1 | 2 | 3 | Total

0 | 0.04 | 0.02 | 0.01 | 0.03 | 0.11

1 | 0.01 | 0.03 | 0.05 | 0.04 | 0.13

2 | 0.02 | 0.01 | 0.04 | 0.69 | 0.76

T | 0.07 | 0.07 | 0.10 | 0.76 | 1.00

b) Probability that at least one waiter and at least one chef show up on any given day = P(X≥1 n Y≥1)

This probability = P(X=1 n Y=1) + P(X=1 n Y=2) + P(X=2 n Y=1) + P(X=2 n Y=2) + P(X=3 n Y=1) + P(X=3 n Y=2)

= 0.03 + 0.01 + 0.05 + 0.04 + 0.04 + 0.69

= 0.86

c) Probability that more chefs show up than waiters on any given day = P(X n Y) where (Y > X)

This probability = P(X=0 n Y=1) + P(X=0 n Y=2) + P(X=1 n Y=2)

= 0.01 + 0.02 + 0.01

= 0.04

d) Probability that more than three total staff (waiters and chefs) will show up on any given day = P(X n Y) where (X + Y) > 3

This probability = P(X=2 n Y=2) + P(X=3 n Y=1) + P(X=3 n Y=2)

= 0.04 + 0.04 + 0.69

= 0.77

Hope this Helps!!!

answer: .64

step-by-step explanation

set up the formula for standard deviation for this set of numbers, then simplify

"it's not the heat, it's the humidity".   that's a partly valid phrase you may have heard in the summer, but it's actually both.   the heat index, also known as the apparent temperature, is what the temperature feels like to the human body when relative humidity is combined with the air temperature.   this has important considerations for the human body's comfort.   when the body gets too hot, it begins to perspire or sweat to cool itself off.   if the perspiration is not able to evaporate, the body cannot regulate its temperature.   evaporation is a cooling process.   when perspiration is evaporated off the body, it effectively reduces the body's temperature.   when the atmospheric moisture content (i.e. relative humidity) is high, the rate of perspiration from the body decreases.   in other words, the human body feels warmer in humid conditions.   the opposite is true when the relative humidity decreases because the rate of perspiration increases.   the body actually feels cooler in arid conditions.   there is direct relationship between the air temperature and relative humidity and the heat index, meaning as the air temperature and relative humidity increase (decrease), the heat index increases (decreases).

step-by-step explanation: