A restaurant manager tracks the number of people in every party to sit at a specific table every day for a week, and then compiles the results into a probability distribution as shown in the table: Number of People , X 12346 Relative Frequency P(X) 0.05 0.46 0.18 0.22 0.06 0.03
This probability distribution represents the likelihood of a certain number of people sitting at the specific table in a day. The random variable X is the number of people in a party, and its possible values are 1, 2, 3, 4, 6, or 0.
The relative frequency column represents the proportion of days in the week where a certain number of people sat at the table. For example, there is a 0.46 probability that a party of 2 people will sit at the table on any given day of the week.
Note that the sum of all the probabilities must equal 1, since there must always be some number of people sitting at the table.
To determine the expected number of people in a party sitting at the specific table, we need to multiply each value of X (number of people) by its corresponding probability P(X), and then sum up the products.
The calculation can be done using the formula for expected value (also known as the mean or average):
E(X) = Σ(X * P(X))
Using the probability distribution table given, we have:
Σ(X * P(X)) = (1 * 0.05) + (2 * 0.46) + (3 * 0.18) + (4 * 0.22) + (6 * 0.06) + (0 * 0.03)
Let's calculate the expected value:
E(X) = (1 * 0.05) + (2 * 0.46) + (3 * 0.18) + (4 * 0.22) + (6 * 0.06) + (0 * 0.03)
= 0.05 + 0.92 + 0.54 + 0.88 + 0.36 + 0
= 2.75
Therefore, the expected number of people in a party sitting at the specific table is 2.75.
To find the expected value or mean number of people in a party at the specific table, you need to multiply each number of people by its respective relative frequency and then sum up the results.
Let's calculate it step by step:
1. Multiply each number of people by its relative frequency:
- 1 * 0.05 = 0.05
- 2 * 0.46 = 0.92
- 3 * 0.18 = 0.54
- 4 * 0.22 = 0.88
- 6 * 0.06 = 0.36
2. Sum up the results:
0.05 + 0.92 + 0.54 + 0.88 + 0.36 = 2.75
So, the expected value or mean number of people in a party at the specific table is 2.75.