In the past year, Jack's Pizaa Resturant grossed more than $1500 a day for about 83% of its business days. Suppose 8 days are randomly selected, approximate the probablity that exactly 6 days will gross more than $1500 in a day.
If the events are independent, the probability of both/all events occurring is determined by multiplying the probabilities of the individual events.
.83^6 * (1-.83)^2 = ?
To approximate the probability, we can use the binomial probability formula:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)
Where:
P(X = k) is the probability of exactly k successes
n is the total number of trials (days selected)
k is the number of successes (days grossing more than $1500)
p is the probability of success in a single trial (probability of grossing more than $1500 on a day)
(1 - p) is the probability of failure in a single trial
First, we need to determine the values for the variables in the formula.
n = 8 (8 days are randomly selected)
k = 6 (exactly 6 days will gross more than $1500)
p = 0.83 (83% of business days gross more than $1500, so the probability of grossing more than $1500 on a day is 0.83)
Now we can substitute these values into the formula:
P(X = 6) = C(8, 6) * 0.83^6 * (1 - 0.83)^(8 - 6)
To calculate C(8, 6), we use the combination formula:
C(8, 6) = 8! / (6! * (8 - 6)!)
Let's simplify the calculations step by step:
C(8, 6) = 8! / (6! * 2!) = (8 * 7 * 6!) / (6! * 2 * 1) = 8 * 7 / (2 * 1) = 28
P(X = 6) = 28 * 0.83^6 * (1 - 0.83)^(8 - 6)
Now we can plug in the values and calculate the probability:
P(X = 6) = 28 * 0.83^6 * (1 - 0.83)^2
P(X = 6) ≈ 28 * 0.310171 * 0.0289 ≈ 0.2416
Therefore, the approximate probability that exactly 6 days will gross more than $1500 in a randomly selected 8-day period is approximately 0.2416, or 24.16%.