A company sells 14 types of crackers that they label varieties 1 through 14, based on spice level. What is the probability that the purchase results in a selection of a cracker with an odd number, or a number greater than 11?

The required probability is

Step-by-step explanation:

Probability: Rule of Addition

The probability that Event A or Event B occurs is equal to the probability that Event A occurs plus the probability that Event B occurs minus the probability that both Events A and B occur. It can be calculated as follows:

P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

There are 14 types of crackers the company sells, numbered from 1 to 14. The sample space for purchasing one cracker is:

There are n=14 possible choices.

The favorable cases for a cracker with an odd number are:

A = {1,3,5,7,9,11,13}

There are 7 favorable cases.

The favorable cases for a cracker with a number greater than 11 are:

B = {12,13,14}

There are 3 favorable cases.

The cases which are common to both events are:

A ∩ B= {13}

There is 1 case.

The probability of each event is:

Therefore:

Adding the fractions:

The required probability is

The only equation that has the vertex up the x-axis and, consequently, does not touch the x-axis is the first one: y = 3x^2 -2x+1.

Then, that is the answer.

To verify that you can calculate the discriminant, b^2 - 4(a)(c), for each equation and use these facts:

If b^2 - 4(a)(c) = 0, there is only one real root (the graph touches the x-axis in one point)

If b^2 - 4ac > 0, there are two real roots (the graph touches the x-axis in two different points)

If b2 - 4ac < 0, there are no real roots (the graph does not touch the x-axis). This is the case for y = 3x^2 - 2x + 1. (-2)^2 -4(3)(1) = 4 - 12 = -8 < 0 => not real roots.