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Chapter 7: Problem 30

What is the probability that a player of a lottery wins the prize offered for correctly choosing five (but not six) numbers out of six integers chosen at random from the integers between 1 and \(40,\) inclusive?

Short Answer

Expert verified
The probability is \( \frac{204}{3,838,380} \).

Step by step solution

01

- Total possible outcomes

Determine the total number of ways to choose 6 numbers out of 40. This can be calculated using the combination formula: \[ \binom{40}{6} = \frac{40!}{6!(40-6)!} \]
02

- Favorable outcomes for choosing 5 correct numbers

There is only one way to pick the 5 correct numbers and there are 34 remaining numbers to choose the incorrect 6th number from. \[ \binom{34}{1} = 34 \]
03

- Calculate probability

Divide the number of favorable outcomes by the total number of possible outcomes: \[ P = \frac{\binom{34}{1}}{\binom{40}{6}} = \frac{34}{\frac{40!}{6!34!}} = \frac{34 \times 6! \times 34!}{40!} \]
04

- Simplify the expression

Calculate the simplified value of the probability: \[ P = \frac{34 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{40 \times 39 \times 38 \times 37 \times 36 \times 35} \] Simplify this fraction step-by-step.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinatorics
Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns in sets. It's super helpful when trying to figure out the number of possible outcomes in a given situation. In the world of combinatorics, you'll see a lot of formulas that make these calculations much easier. For example, when determining how many ways you can choose a certain number of items from a larger set, you use the combination formula. This formula helps you calculate the number of possible selections without considering the order of the items.
Probability
Probability is the branch of mathematics that deals with the likelihood of an event occurring. It's all about measuring how likely something is to happen. To calculate probability, you need to know two things: the number of favorable outcomes and the total number of possible outcomes. The formula for probability is simple:
\[ P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \]
In the context of our lottery problem, we're looking at the probability of picking 5 out of 6 correct numbers out of a pool of 40. So, we calculate the favorable outcomes (choosing 5 right numbers and 1 wrong number) and compare that to the total number of ways to pick 6 numbers from a pool of 40.
Combination Formula
The combination formula is a tool used in combinatorics to find the number of ways to choose a subset of items from a larger set, without regard to the order of the items. The formula is:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n \) is the total number of items to choose from, \( r \) is the number of items you want to choose, and \( ! \) (factorial) represents the product of all positive integers up to that number. For example, in our lottery problem, we used \( \binom{40}{6} \) to determine the number of ways to choose 6 numbers from a pool of 40. This was followed by \( \binom{34}{1} \) to find the number of ways to select the incorrect 6th number from the remaining 34 numbers.
Lottery Odds
Lottery odds are all about figuring out your chances of winning a particular lottery game. Understanding these odds involves combinatorics and probability. For instance, in our exercise, we calculate the odds of picking 5 out of 6 correct numbers in a lottery.
The process includes:
  • Calculating the total number of possible outcomes using the combination formula \( \binom{40}{6} \)
  • Determining the number of favorable outcomes where you pick the 5 correct numbers and 1 incorrect number \( \binom{34}{1} \)
  • Using the probability formula to find your chances
These steps give you a clear picture of just how likely (or unlikely) it is to win the lottery based on specific criteria. Remember, lottery odds can be quite daunting due to the large number of possible combinations, but with the right calculations, you can understand exactly what you're up against.

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