In the next preview assignment and in the next class, you will need to compute probabilities that involve exponents, and you will need to interpret statements of inequality and use those to find probabilities.
How Many 1’s?
When events [latex]A[/latex] and [latex]B[/latex] are independent, then [latex]P(A\mbox{ and }B) = P(A)∙P(B)[/latex].
When using this rule, there are situations in which the probabilities of some events are the same, and that’s when exponents come into play.
Exponent Review: [latex]p^{x}[/latex] means that [latex]p[/latex] is multiplied by itself [latex]x[/latex] times. For example, [latex](0.5)^{2} = (0.5) \bullet (0.5) = 0.25, \mbox{ and } (0.5)^{3} = (0.5) \bullet (0.5) \bullet (0.5) = 0.125[/latex].
Recall that multiplication is commutative, which means that the order we multiply numbers in does not matter when multiplication is the only operation present. For example:
[latex](0.25) \bullet (0.5) \bullet (0.25) = (0.25) \bullet (0.25) \bullet (0.5) = (0.25)^{2}(0.5)[/latex]

Example: Suppose we are rolling a fair, 6-sided die 4 times. The probability that the first roll is not a 6, the second roll is a 6, and the final 2 rolls are also not 6 is:
[latex]P(not~6~AND~6~AND~not~6~AND~not~6) = P(not~6) * P(6) * P(not~6) * P(not~6) = P(6) \bullet (P(not~6))^{3} = (\frac{1}{6}) \bullet (\frac{5}{6})^{3} \approx 0.0965[/latex]

In the next preview assignment, you will be considering probability questions where each trial of an experiment has exactly two outcomes (which we call “success” and “failure”), and we want to count how many of the trials result in successes.

In the context of our spinner question, we might want to count how many times the spinner lands on the 1 section if we spin the spinner 10 times. We will also consider questions like, “What is the probability that in at least 6 out of 10 spin attempts, the arrow lands on the 1 section?” For these types of questions, we will need inequality expressions.

Inequality Review – There are four inequality symbols:

• [latex]<[/latex] (less than): [latex]a
• [latex]\leq[/latex] (less than or equal to): [latex]a \leq b[/latex] means that “a is less than or equal to b.” For example, we can write [latex]2 \leq 5[/latex] and [latex]5 \leq 5[/latex] because the symbol indicates that there are two possibilities: the number on the left is either less than 5 or the number on the left equals 5.
• [latex]>[/latex] (greater than): [latex]a>b[/latex] means that “a is greater than b.” For example, we write [latex]5 > 2[/latex] because 5 is greater than 2.
• [latex]\geq[/latex] (greater than or equal to): [latex]a \geq b[/latex] means that “a is greater than or equal to b.” For example, we can write [latex]5 \geq 2[/latex] and [latex]5 \geq 5[/latex] because the symbol indicates that there are two possibilities: the number on the left is either greater than 5 or the number on the left equals 5.

One way to help remember how these symbols work is that the pointy end (the smaller side) of the symbol always points to the smaller number, while the open side (the bigger side) of the symbol always opens to the bigger number. (A fun way to remember it is to think of the symbol as a mouth that wants to eat the biggest number!)

An important part of answering probability questions is translating the question into a mathematical expression so that you know what probability you’re trying to find.

Example: As above, let’s imagine we are spinning a spinner with 4 equally-sized sections 10 times. Let the random variable [latex]X[/latex]be the number of times that we land on the 1 in our 10 spin attempts. Then we can describe the probability that we land on 1 on more than 5 spins as [latex]P(X> 5)[/latex] since we want the number of spins where the arrow lands on 1 to be more than 5.

Sometimes, a description of an inequality can be a little trickier to interpret.

• “At least x” means x or more. For example, “at least 3” means 3 or more.
• “Up to x” means less than or equal to x. For example, “up to 2” means 2 or less.
• “At most x” means less than or equal to x. For example, “at most 2” means 2 or less.
• “No more than x” means less than or equal to x. For example, “no more than 2” means 2 or less.