Question

If a person is given the choice of an integer from 0 to 9 , is it more likely that he or she will choose an integer near the middle of the sequence than one at either end? a. If the integers are equally likely to be chosen, find the probability distribution for \(x\), the number chosen. b. What is the probability that a person will choose a \(4,5,\) or \(6 ?\) c. What is the probability that a person will not choose a \(4,5,\) or \(6 ?\)

Short answer

Answer: The probability of a person choosing 4, 5, or 6 is 3/10, and the probability of a person not choosing 4, 5, or 6 is 7/10.

Step-by-step solution

Finding the probability distribution for x

Since there are 10 integers (0,1,2,3,4,5,6,7,8,9) and each integer is equally likely to be chosen, the probability of choosing any single integer is 1 divided by the total number of integers which is 10. Therefore, the probability distribution for x is P(x) = 1/10 for each integer between 0 and 9.

Calculate the probability of choosing 4, 5, or 6

To find the probability of choosing a 4, 5, or 6, we will add the probabilities of choosing each of these integers. Since the probability of choosing each number is 1/10, the probability of choosing either 4, 5, or 6 is: P(4, 5, or 6) = P(4) + P(5) + P(6) = (1/10) + (1/10) + (1/10) = 3/10

Calculate the probability of not choosing 4, 5, or 6

Since the probability of choosing 4, 5, or 6 is equal to 3/10 and the total probability must be equal to 1, the probability of not choosing 4, 5, or 6 is given by: P(not 4, 5, or 6) = 1 - P(4, 5, or 6) = 1 - 3/10 = 7/10 So, the probability of not choosing 4, 5, or 6 is 7/10.

Key concepts

Uniform Distribution

A uniform distribution is a probability distribution where each outcome is equally likely. In simpler terms, imagine you have a perfectly fair dice or a balanced spinner. Every possible outcome has the same chance of occurring.
In the context of this exercise, the integers from 0 to 9 have a uniform probability distribution. This means each number is equally likely to be chosen. If you are picking one number at random, the likelihood (probability) that you choose any specific number, say 7, is the same as picking any other number, such as 2.

• Each integer between 0 and 9 has a probability of being selected of \( \frac{1}{10} \) because there are 10 equally likely integers.

This is a classic example of uniform distribution in discrete random variables, where the probabilities are evened out across all possible choices.

Elementary Probability

Elementary probability is the foundation of all probability concepts. It involves understanding basic probability theory and simple calculations. The essence of elementary probability lies in how likely an event is to happen.
In the given exercise, you calculate the likelihood of a specific set of outcomes, for example, the chance a person will choose the numbers 4, 5, or 6 when picking randomly from 0 to 9. Since each number is equally likely (as explained in the uniform distribution section), the probability for each number is \( \frac{1}{10} \).

• For multiple events like choosing either 4, 5, or 6, you add their individual probabilities:
• \( P(4) + P(5) + P(6) = \frac{1}{10} + \frac{1}{10} + \frac{1}{10} = \frac{3}{10} \)

Using elementary probability, you calculate compound events by summing the probabilities of individual events if events are mutually exclusive, like choosing different numbers.

Discrete Probability

Discrete probability examines the likelihood of outcomes in scenarios where results can be counted as distinct items.
Think of rolling a dice, where each side of the dice represents a discrete number. Unlike continuous probability, where outcomes lie in a continuum or range, discrete outcomes are countable.

• In this problem, the numbers 0 through 9 are discrete outcomes. You can list and separate each by itself.
• Since the choice is made randomly and each number stands alone (independent of others), you find individual probabilities for each distinct number, such as choosing 4.

Thus, the probability of specific events can be computed directly, such as not choosing a 4, 5, or 6.
The sum of all probabilities in a discrete distribution should always be 1, signifying the certainty of choosing a number from the list. Hence, the probability of not choosing one event can be calculated by subtracting the probability of that event from 1, which represents total certainty. For example, \( P(\text{not 4, 5, or 6}) = 1 - \frac{3}{10} = \frac{7}{10} \). This is a key aspect of discrete probability distribution.