In the town of Niceville, a local lottery is held once a year for the residents. Lottery tickets cost 50 cents apiece. Each lottery ticket contains four numbers, from 1 to 20. The lottery machine will select six numbers from 1 to 20. A winning ticket, worth $100, will contain four of the six numbers chosen by the machine.
a)How much money would a person need to spend to buy every possible combination of four numbers?
b)What is the probability, to the nearest ten-thousandth, that a person will select four correct numbers?
c)To the nearest ten-thousadnth, what is the probability that a ticket has NO correct numbers?
Can you please explain how you got the answer also? Thanks so much~
a) To calculate the amount of money needed to buy every possible combination of four numbers, we need to determine the total number of combinations. The formula for calculating a combination is nCr = n! / (r!(n-r)!), where n is the total number of items to choose from and r is the number of items being chosen.
In this case, we have 20 numbers to choose from (1 to 20) and we need to choose 4 numbers. Using the combination formula, we can calculate the total number of combinations as follows:
nCr = 20! / (4!(20-4)!)
= (20 * 19 * 18 * 17!) / (4 * 3 * 2 * 1 * 16!)
Note: n! means to multiply all positive integers from 1 to n together.
Simplifying this expression, we find:
nCr = (20 * 19 * 18 * 17) / (4 * 3 * 2 * 1)
= 4845
So, there are a total of 4,845 possible combinations of four numbers.
Since each combination costs 50 cents, the equation for determining the total cost is:
Total cost = Number of combinations * Cost per combination
= 4,845 * $0.50
= $2,422.50
Therefore, a person would need to spend $2,422.50 to buy every possible combination of four numbers.
b) To calculate the probability of selecting four correct numbers, we need to determine the number of favorable outcomes (the number of ways to choose four numbers correctly) and the total number of possible outcomes (the number of possible combinations of four numbers).
The number of favorable outcomes is simply 1, as there is only one winning combination.
The total number of possible outcomes is the same as the total number of combinations, which we calculated as 4,845 in part a).
Therefore, the probability is:
Probability = Favorable outcomes / Total possible outcomes
= 1 / 4,845
≈ 0.000206
So, the probability of selecting four correct numbers is approximately 0.000206 or 0.0206%.
c) To calculate the probability of having no correct numbers, we need to determine the number of favorable outcomes and the total number of possible outcomes.
In this case, the number of favorable outcomes is the number of ways to choose four wrong numbers out of the six numbers selected by the machine.
We can calculate this using the combination formula with n=14 (20 - 6) and r=4:
Number of favorable outcomes = 14C4 = 14! / (4!(14-4)!)
= (14 * 13 * 12 * 11) / (4 * 3 * 2 * 1)
= 1001
The total number of possible outcomes is the same as the total number of combinations, which we calculated as 4,845 in part a).
Therefore, the probability is:
Probability = Favorable outcomes / Total possible outcomes
= 1001 / 4845
≈ 0.2067
So, the probability of a ticket having no correct numbers is approximately 0.2067 or 20.67%.
a) To determine the amount of money a person would need to spend to buy every possible combination of four numbers, we first need to find the total number of combinations. Since there are 20 numbers to choose from and we need to select 4 numbers, we can use the combination formula:
nCr = n! / (r!(n-r)!)
In this case, n = 20 (total numbers) and r = 4 (numbers to be selected).
20C4 = 20! / (4!(20-4)!) = 4845
So, there are a total of 4845 possible combinations of four numbers.
To calculate the total cost, we need to multiply the number of combinations by the price of a single ticket.
Cost = 4845 * $0.50 = $2422.50
Therefore, a person would need to spend $2422.50 to buy every possible combination of four numbers.
b) The probability of selecting four correct numbers can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
The total number of possible outcomes is the same as the number of combinations we calculated in part (a): 4845.
To find the number of favorable outcomes, we need to consider the fact that there are 6 winning numbers selected by the lottery machine. We need to choose 4 of those numbers correctly.
The number of favorable outcomes can be calculated using the combination formula:
6C4 = 6! / (4!(6-4)!) = 15
So, there are 15 favorable outcomes.
Probability = Favorable outcomes / Total outcomes = 15 / 4845 ≈ 0.00309
Therefore, the probability, to the nearest ten-thousandth, that a person will select four correct numbers is approximately 0.0031.
c) To calculate the probability that a ticket has no correct numbers, we need to find the number of favorable outcomes where none of the chosen numbers match the winning numbers.
In this case, we want to select 4 numbers that are not among the 6 winning numbers chosen by the machine.
The number of favorable outcomes can be calculated using the combination formula:
(20-6)C4 = 14C4 = 14! / (4!(14-4)!) = 1001
So, there are 1001 favorable outcomes.
Probability = Favorable outcomes / Total outcomes = 1001 / 4845 ≈ 0.2068
Therefore, the probability, to the nearest ten-thousandth, that a ticket has no correct numbers is approximately 0.2068.