Problem 73 - Algebra and Trigonometry

Question

Lotto America is a multistate lottery in which 5 red balls from a drum with 52 balls and 1 star ball from a drum with 10 balls are selected. For a $$\$ 1$$ ticket, players get one chance at winning the grand prize by matching all 6 numbers. What is the probability of selecting the winning numbers on a $$\$ 1$$ play?

Step-by-Step Solution

Answer

The probability of winning is \( \frac{1}{25,989,600} \).

Step 1: Understanding the Problem

To find the probability of winning, we need to determine the total number of possible combinations for selecting 5 red balls from 52 and 1 star ball from 10, and then calculate the probability of selecting the exact combination.

Step 2: Calculate the Total Combinations for Red Balls

Use the combination formula to determine the number of ways to choose 5 balls from 52. The formula for combinations is given by \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Here, we need to calculate \( \binom{52}{5} \): \[ \binom{52}{5} = \frac{52!}{5!(52-5)!} = \frac{52!}{5! \times 47!} \]

Step 3: Calculate the Total Combinations for Star Ball

There are 10 possible star balls, so there are 10 combinations for choosing 1 star ball from 10.

Step 4: Multiply the Combinations

Multiply the number of combinations for the red balls by the number of combinations for the star ball to get the total number of possible combinations: \[ \binom{52}{5} \times 10 \]

Step 5: Calculate the Probability

The probability of winning is the reciprocal of the total number of possible combinations. So,\[ \text{Probability} = \frac{1}{\binom{52}{5} \times 10} = \frac{1}{2,598,960 \times 10} = \frac{1}{25,989,600} \]

Key Concepts

Combinatorics in Lottery Games

In lottery games, combinatorics plays a vital role. It helps us understand how to count combinations.
Combinations are different from permutations; order does not matter in combinations.
When calculating combinations, we use the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] Here, \( n \) stands for the total number of items, and \( k \) stands for the number of items to choose.
In the lotto exercise, we find the combination for 5 red balls from a total of 52 balls.
This is because any 5 chosen balls are regarded equally, and their order does not matter.
Using combinatorics helps us get a consistent and accurate count simply by plugging values into the formula.

Probability Calculation in Lotteries

Probability calculation determines the likelihood of a specific outcome happening.
In the lotto example, we are interested in the probability of winning.
This requires knowing both the total number of possible combinations and the exact combination you need to win.
First, we calculate the total number of combinations for 5 red balls from 52 as \( \binom{52}{5} \).
Then, we find the number of combinations for 1 star ball from 10.
Finally, we multiply these two values to get the total number of ways to pick the winning combination.
The probability of winning is the reciprocal of this total number.
So, the probability of winning is calculated as: \[ \text{Probability} = \frac{1}{\binom{52}{5} \times 10} = \frac{1}{2,598,960 \times 10} = \frac{1}{25,989,600} \]
This shows that winning the lotto is extremely rare.

Understanding Factorials

Factorials are fundamental in combinatorics.
A factorial of a non-negative integer, denoted by \( n! \), is the product of all positive integers less than or equal to \( n \).
For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorial values grow extremely quickly, even for small numbers.
Factorials are used in the combination formula to handle the repeated items and ordering possibilities correctly.
In our lotto example, calculating \( \binom{52}{5} \) involves computing \( 52! \), \( 5! \), and \( 47! \).
With these values, the formula simplifies the complex calculation needed to determine combinations.
Learning and understanding factorials not only helps in probability calculations but also in various other areas of mathematics and real-life problems.