Probability of Exactly One Value

For continuous distributions:

• The probability of being exactly at one value is zero.
• Ie: [latex]P(x= X)=0[/latex]

Because of this:

• [latex]P(x\ge X)=P(x= X)+P(x\gt X) = 0+P(x\gt X) = P(x\gt X)[/latex]
• [latex]P(x\le X)= P(x= X)+P(x\lt X) = 0+P(x\lt X) = P(x\lt X)[/latex]

At Least or More Than

For the probability of at least or more than [latex]X[/latex], ie: [latex]P(x\ge X)[/latex] or [latex]P(x\gt X)[/latex]

• [latex]x_1 = X[/latex] (the lowest value in the [latex]x[/latex]-range)
• [latex]x_2 = b[/latex] (the highest possible [latex]x[/latex]-value)

This gives [latex]P(x_1 \le x \le x_2) =  \frac{x_2 - x_1}{b-a} = \frac{b-X}{b-a}[/latex].

At Most or Less Than

For the probability of at most or less than [latex]X[/latex], ie: [latex]P(x\le X)[/latex] or [latex]P(x\lt X)[/latex]:

• [latex]x_1 = a[/latex] (the lowest possible [latex]x[/latex]-value)
• [latex]x_2 = X[/latex] (the highest value in the [latex]x[/latex]-range)

This gives [latex]P(x_1 \le x \le x_2) =  \frac{x_2 - x_1}{b-a} = \frac{X-a}{b-a}[/latex]

Travel Times (At Least Exercise)

Let us look at calculating the probabilities of at least or more than a certain [latex]x[/latex] value.

Example 35.1.1

Problem Setup: During the morning commute, the time it takes to drive to BCIT follows a uniform distribution and is between 30 and 55 minutes. You slept in this morning and woke up 45 minutes before your class is about to begin.

Question: If it only takes you 5 minutes to get ready, what are the odds that you will be late for class?

You Try: First, let’s fill in the values for [latex]a[/latex], [latex]b[/latex], [latex]x_1[/latex], and [latex]x_2[/latex]:

Solution: Next, let’s plug them into the equation:

\[ P(\text{late for class}) = P(x \ge 40) = \frac{x_2-x_1}{b-a}=\frac{55-40}{55-30} = 0.6 \]

Conclusion: There is a 60% chance that you will be late for class this morning.

Hints for Exercise: Click below to reveal explanations for the above exercise.

Travel Times Example Continued (Less Than Video)

Let us now change up the problem slightly and show the solution in a video (see below).

Example 35.1.2

Problem Setup: Let us continue on with the previous example:

• You have 40 minutes to get to class.
• Your commute times are between 30 and 55 minutes.
• The times follow a uniform distribution.

Question: What is the probability of being on time or early for class?

Solution: Click here to download the written solutions. Also, see the video below:

Key Takeaways (EXERCISE)

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