Five numbers are to be picked, without repetition, from 44
numbers to determine the winner of the Fortune Five
game in the state lottery. If the order of the numbers is
insignificant, how many different ways can a winning
quintuple be selected? What is the probability of winning?
To determine the number of different ways a winning quintuple can be selected, we can use the concept of combinations. A combination refers to the selection of items from a larger set without considering the order in which they are chosen.
In this scenario, we have 44 numbers to choose from, and we need to select 5 numbers without repetition. The formula to calculate the number of combinations is:
C(n, r) = n! / (r! * (n - r)!)
Where:
- n is the total number of items in the set (44 numbers)
- r is the number of items we want to select (5 numbers)
- ! denotes the factorial function, which is the product of all positive integers up to a given number
Using this formula:
C(44, 5) = 44! / (5! * (44-5)!)
= (44 * 43 * 42 * 41 * 40) / (5 * 4 * 3 * 2 * 1)
= 1,086,008
There are 1,086,008 different ways a winning quintuple can be selected.
Now, let's calculate the probability of winning. To find the probability, we need to divide the number of favorable outcomes (winning quintuples) by the total number of possible outcomes.
The total number of possible outcomes is the number of ways we can select any 5 numbers from the 44 available, which is the same as the number of combinations we just calculated: 1,086,008.
Assuming there is only one winning quintuple, the number of favorable outcomes is 1.
Therefore, the probability of winning is:
P(winning) = Number of favorable outcomes / Total number of possible outcomes
= 1 / 1,086,008
The probability of winning is approximately 0.00000092, or 0.000092%.