Cellular Phone Sales The probability that a cellular phone company kiosk sells X number of new phone contracts per day is shown below. Find the mean, variance, and standard deviation for this probability distribution.
X
4
5
6
8
10
P (X)
0.4
0.3
0.1
0.15
0.05
What is the probability that they will sell 6 or more contracts three days in a row?
The probability that they will sell 6 or more contracts three days in a row is 0.001 (0.1 x 0.1 x 0.1).
Mean: 6.2
Variance: 4.68
Standard Deviation: 2.16
To find the mean, variance, and standard deviation for the given probability distribution, you can use the following formulas:
Mean (μ) = Σ(X * P(X))
Variance (σ^2) = Σ((X - μ)^2 * P(X))
Standard Deviation (σ) = √σ^2
Let's calculate these values step by step:
Mean:
Multiply each value of X by its corresponding probability and then sum them up:
4 * 0.4 + 5 * 0.3 + 6 * 0.1 + 8 * 0.15 + 10 * 0.05 = 1.6 + 1.5 + 0.6 + 1.2 + 0.5 = 5.4
Variance:
First, subtract the mean (5.4) from each value of X, square the result, multiply by the corresponding probability, and then sum them up:
((4 - 5.4)^2 * 0.4) + ((5 - 5.4)^2 * 0.3) + ((6 - 5.4)^2 * 0.1) + ((8 - 5.4)^2 * 0.15) + ((10 - 5.4)^2 * 0.05) = 2.56 * 0.4 + 0.16 * 0.3 + 0.36 * 0.1 + 5.76 * 0.15 + 18.36 * 0.05 = 1.024 + 0.048 + 0.036 + 0.864 + 0.918 = 2.89
Standard Deviation:
Take the square root of the variance:
√2.89 ≈ 1.70
Now, let's calculate the probability that they will sell 6 or more contracts three days in a row. Since the three days are independent events, you can multiply the probabilities of each day.
The probability of selling 6 or more contracts on any given day is the sum of the probabilities for each value of X equal to or greater than 6:
P(X ≥ 6) = P(X = 6) + P(X = 8) + P(X = 10) = 0.1 + 0.15 + 0.05 = 0.3
To find the probability for three days in a row, multiply this probability by itself three times:
P(6 or more contracts for three days in a row) = (0.3)^3 = 0.027 or 2.7%
Therefore, the probability that they will sell 6 or more contracts three days in a row is 0.027 (or 2.7%).
To find the probability that they will sell 6 or more contracts three days in a row, we need to multiply the probabilities of each individual day.
The probability of selling 6 or more contracts on any given day is the sum of the probabilities for X = 6, 8, and 10.
P(X = 6) = 0.1
P(X = 8) = 0.15
P(X = 10) = 0.05
Now, we can calculate the probability of selling 6 or more contracts three days in a row by multiplying the probabilities for each day.
P(6 or more contracts on day 1) = P(X = 6) + P(X = 8) + P(X = 10) = 0.1 + 0.15 + 0.05 = 0.3
Therefore, the probability that they will sell 6 or more contracts on the first day is 0.3.
Similarly, the probability of selling 6 or more contracts on the second day is also 0.3.
To find the probability of three days in a row, we multiply the probabilities of each individual day.
P(6 or more contracts for three days in a row) = P(6 or more contracts on day 1) × P(6 or more contracts on day 2) × P(6 or more contracts on day 3) = 0.3 × 0.3 × 0.3 = 0.027
Therefore, the probability that they will sell 6 or more contracts three days in a row is 0.027.