Customers at a gas station pay with a credit card (A), debit card (B), or cash (C). Assume that successive customers make independent choices with P(A) = 0.6, P(B) = 0.3, and P(C) = 0.1.

(a) Among the next 100 customers, what are the mean and variance of the number who pay with a debit card?

Mean = 20

Variance = 16

Step-by-step explanation:

Solution:-

- Let X be the number of customers paying with a debit card.  X has the binomial distribution with n = 100 trials and success probability p = 0.2  

- In general, if X has the binomial distribution with (n) trials and a success probability of (p) then:

                 P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)

                 for values of x = 0, 1, 2, ..., n  

                 P[X = x] = 0 for any other value of x.

- The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.  

- Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.

                  X ~ Binomial( n = 100 , p = 0.2 )

- The mean of the binomial distribution is n * p = 20  

- The variance of the binomial distribution is n * p * (1 - p) = 16  

x = ±sqrt(10)

step-by-step explanation:

5 - 2x² = -15

subtract 5 from each side

5-5 - 2x² = -15-5

-2x^2   = -20

divide by -2 on each side

-2x^2   /-2= -20/-2

x^2 =10

take the square root on each side

sqrt(x^2 )=±sqrt(10)

x = ±sqrt(10)

check:

5 - 2x² = -15

+sqrt(10)

5 - 2(sqrt(10))² = -15

5 -2(10) = -15

5 -20 = -15

-15 = -15

-sqrt(10)

5 - 2(-sqrt(10))² = -15

5 -2(10) = -15

5 -20 = -15

-15 = -15