Question 1165742: In a lottery game, a player picks 8 numbers from 1 to 48. If 6 of those 8 numbers match those drawn, the player wins third prize. Let's walk through the steps to determine the probability of winning third prize.
In how many ways can 6 winning numbers be chosen from the possible 8 numbers?
In how many ways can 2 non-winning numbers be chosen from the pool of all non-winning numbers?
The number of favorable outcomes would be to multiply the above two answers together, since we want 6 winning numbers and 2 non-winning numbers. What is the number of favorable outcomes?
In how many ways can you pick any 8 numbers from the pool of 48 numbers? This is your total outcomes.
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What is the probability of winning third prize?
(Round to 5 decimal places.)
Answer by CPhill(2189)   (Show Source):
You can put this solution on YOUR website! This is a classic probability problem that uses combinations. Let's break down the steps to find the probability of winning the third prize.
The game parameters are:
* Total numbers available: **48**
* Numbers drawn (Winning numbers): **8**
* Numbers a player picks: **8**
* Winning Condition (Third Prize): Matching exactly **6** of the 8 drawn numbers.
This means a favorable outcome requires matching 6 winning numbers and 2 non-winning numbers.
## 1. Calculating Favorable Outcomes
First, we need to determine the number of ways to pick 6 winning numbers and 2 non-winning numbers.
### A. Ways to Choose 6 Winning Numbers
You need to choose **6** numbers from the **8** winning numbers drawn.
$$\binom{8}{6} = \frac{8!}{6!(8-6)!} = \frac{8 \times 7}{2 \times 1} = \mathbf{28}$$
* **In how many ways can 6 winning numbers be chosen from the possible 8 numbers?** $\mathbf{28}$ ways.
### B. Ways to Choose 2 Non-Winning Numbers
The total pool is 48 numbers. Since 8 are winning numbers, the number of non-winning numbers is $48 - 8 = 40$. You need to choose **2** numbers from these **40** non-winning numbers.
$$\binom{40}{2} = \frac{40!}{2!(40-2)!} = \frac{40 \times 39}{2 \times 1} = \mathbf{780}$$
* **In how many ways can 2 non-winning numbers be chosen from the pool of all non-winning numbers?** $\mathbf{780}$ ways.
### C. Total Favorable Outcomes
The total number of favorable outcomes is the product of the ways to choose the winning and non-winning numbers.
$$\text{Favorable Outcomes} = \binom{8}{6} \times \binom{40}{2} = 28 \times 780 = \mathbf{21,840}$$
* **The number of favorable outcomes is $\mathbf{21,840}$.**
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## 2. Calculating Total Possible Outcomes
You pick **8** numbers from the pool of **48** numbers. This represents the total number of possible combinations the player can choose.
$$\text{Total Outcomes} = \binom{48}{8}$$
$$\binom{48}{8} = \frac{48!}{8!(48-8)!} = \frac{48 \times 47 \times 46 \times 45 \times 44 \times 43 \times 42 \times 41}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$$
$$\text{Total Outcomes} = \mathbf{377,370,144}$$
* **In how many ways can you pick any 8 numbers from the pool of 48 numbers?** $\mathbf{377,370,144}$ ways.
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## 3. Probability of Winning Third Prize
The probability is the ratio of favorable outcomes to total possible outcomes.
$$\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{21,840}{377,370,144}$$
$$\text{Probability} \approx 0.000057874...$$
Rounding to 5 decimal places: $\mathbf{0.00006}$
* **What is the probability of winning third prize? (Round to 5 decimal places.)** $\mathbf{0.00006}$
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