A new lottery rule of winning Lotto 6-49 is when all of your numbers match or 5 of your numbers match. Find the probability that you win the lottery.
To find the probability of winning the lottery with the new rule, we need to consider two scenarios:
Scenario 1: All of your numbers match.
Scenario 2: Exactly 5 of your numbers match.
Let's break down each scenario to calculate the probability.
Scenario 1: All of your numbers match
To calculate the probability of getting all your numbers correct, we need to determine the number of ways you can choose all the correct numbers out of the total possible combinations.
In Lotto 6-49, you need to choose 6 numbers from a pool of 49 numbers. The number of possible combinations is given by the formula:
C(n, k) = n! / (k! * (n-k)!),
where
n is the total number of objects (49 in this case),
k is the number of objects to be chosen (6 in this case),
n! represents the factorial of n.
Therefore, the number of possible combinations is:
C(49, 6) = 49! / (6! * (49-6)!) = 13,983,816.
Since you need to match all the numbers, there is only one winning combination. So, the probability of winning in this scenario is:
1 / 13,983,816 ≈ 7.15 x 10^-8 (approximately).
Scenario 2: Exactly 5 of your numbers match
To calculate the probability of getting exactly 5 numbers correct, we need to determine the number of ways you can choose 5 correct numbers out of the 6 numbers you chose.
Similarly, the number of possible combinations is given by the formula:
C(n, k) = n! / (k! * (n-k)!).
In this scenario, you have 6 ways to choose the incorrect number out of 49 numbers. Therefore, the total number of combinations is:
C(6, 5) * C(49, 1) = (6! / (5! * (6-5)!)) * (49! / (1! * (49-1)!)) = 6 * 49 = 294.
So, the probability of winning in scenario 2 is:
294 / 13,983,816 ≈ 2.10 x 10^-5 (approximately).
Finally, to find the overall probability of winning the lottery, we add the probabilities of scenario 1 and scenario 2 together:
7.15 x 10^-8 + 2.10 x 10^-5 ≈ 2.10 x 10^-5 (approximately).
Therefore, the probability of winning the lottery is approximately 2.10 x 10^-5.