Question

A lottery game has players selecting 3 numbers at random from the numbers 1-50. The state picks 20 winning numbers. If the​ player's 3 numbers are selected by the​ state, the player wins. What is the probability that the state picks none of the​ player's 3 ​numbers?

How many ways can the 3 numbers be chosen from the original 50 ​numbers, regardless of whether they are in the winning group or the losing​ group

The probability that the state picks none of the​ player's 3 numbers is

Answer 1

The probability that the state picks none of the player’s 3 numbers in a lottery game is calculated by the combination C(47,20) divided by C(50,20). The number of ways to choose any 3 numbers from 50 is given by the combination C(50,3).

Step-by-step explanation:

Calculating Probability in a Lottery Game

To calculate the probability that the state picks none of the player’s 3 numbers, we have to consider the ways the 20 winning numbers can be picked from the 47 remaining numbers (after excluding the player's 3 selected numbers). The total number of combinations for choosing 3 numbers from 50 is calculated using combinations, denoted as C(n,r), where n is the total number of items, and r is the number of items to choose. In this case, it's C(50,3).

The number of ways to choose 20 winning numbers from the 47 that are not the player’s numbers would be C(47,20). To find the total number of possible outcomes for the 20 winning numbers, regardless of the player's numbers, we use C(50,20).

The probability of the state not picking any of the player's numbers is then the number of ways to choose 20 numbers from 47, divided by the total number of ways to choose 20 numbers from 50:

Number of ways to choose 3 numbers from 50: C(50,3)

Probability state picks none of player's numbers: C(47,20) / C(50,20)