Problem 11 - Thinking Mathematically

Question

Mega Millions is a multi state lottery played in most U.S. states. As of this writing, the top cash prize was \(\$ 656\) million, going to three lucky winners in three states. Players pick five different numbers from 1 to 56 and one number from 1 to 46. Use this information to solve Exercises 7-10. Express all probabilities as fractions. A box contains 25 transistors, 6 of which are defective. If 6 are selected at random, find the probability that a. all are defective. b. none are defective.

Step-by-Step Solution

Answer

a) The probability that all are defective is \(C(6,6) / C(25,6)\)b) The probability that none is defective is \(C(19,6) / C(25,6)\)

Step 1: Calculation for Scenario A

Find the probability that all selected transistors are defective. This involves selecting all 6 defective transistors from the total number of defective transistors. Use the formula for combinations \(C(n, k)\) which is \(C(n, k) = n! / (k!(n-k)!)\). Here \(n\) represents the total number of defective transistors while \(k\) represents the number of transistors to be selected. Find the combination \(C(6,6)\). In the possible set of 6 transistors, find the total combinations \(C(25,6)\). Finally, calculate the probability by dividing \(C(6,6)\) by \(C(25,6)\).

Step 2: Calculation for Scenario B

Find the probability that none of the selected transistors are defective. This involves selecting all 6 transistors from the total number of non-defective transistors. Use the formula for combinations to find \(C(19, 6)\). Calculate the total combinations in the set of 6 transistors as \(C(25, 6)\). Finally, find the probability by dividing \(C(19, 6)\) by \(C(25, 6)\).

Key Concepts

Combinations Formula

The combinations formula is used when you want to select items from a group where the order does not matter. Mathematically, for a set of 'n' items if you want to choose 'k' items, the formula is written as:

\[\begin{equation}C(n, k) = \frac{n!}{k!(n-k)!}\end{equation}\]

The exclamation points in the formula represent the factorial notation (covered in the next section). Using this formula can tell you how many different groups of 'k' items you can make from a larger set of 'n' items. For example, if you have a 25 transistor box and want to know how many ways you can select 6, the combinations formula will provide the answer.

Factorial Notation

Factorial notation is represented by an exclamation point (!) and is a product of all positive integers less than or equal to a number. For any non-negative integer 'n', factorial is defined as:

\[\begin{equation}n! = n \times (n-1) \times (n-2) \times ... \times 1\footnotesize{(n eq 0)}\end{equation}\]

For example, the factorial of 6 (6!) would be calculated as 6 x 5 x 4 x 3 x 2 x 1 = 720. It's an essential concept in probability and combinations because it determines how many ways you can arrange 'n' items. Zero factorial is defined to be 1, which is vital to remember when calculating probabilities and combinations.

Probability as Fractions

In probability, outcomes can be expressed as fractions, which represent the portion of all possible outcomes the event represents. Probabilities as fractions are always between 0 and 1, where 0 means an event is impossible and 1 means it is certain. For example, if there is one desired outcome and four possible outcomes altogether, the probability is written as the fraction 1/4. This form is particularly helpful when you need to clearly see the relationship between the successful outcomes and all possible outcomes, as in the case of selecting defective or non-defective transistors from a box.

Defective Selection Probability

The probability of selecting defective items plays a significant role in quality control and decision-making processes. In probability theory, the chance of picking a certain number of defective items from a batch involves determining the amount of defective items and possible selections. As with the transistor example, if you want to calculate the probability of selecting all defective items, you combine your understanding of combinations and probability fractions. In context, selecting all 6 defective transistors from a box of 25, where 6 are defective, requires finding the single combination of 6 defective (which is one, or \[\begin{equation}C(6,6)\end{equation}\]), and dividing it by the total combinations of selecting any 6 from the 25 (which can be calculated using the combinations formula). This gives you the probability of an all-defective selection in fraction form.