Find the probability of winning a lottery with the following rule.
Select the five winning numbers from 1,2...,23. (in any oder. No repeats)
P(winning)= ?
To find the probability of winning a lottery with the given rule, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of possible outcomes:
Since we are selecting 5 numbers from a range of 1 to 23 without repeats, we can calculate the total number of outcomes using combination formula. In this case, we can use "23 choose 5" represented as C(23, 5).
Number of favorable outcomes:
The number of favorable outcomes is just 1 because there is only one set of winning numbers.
Now, let's calculate the probability of winning the lottery:
P(winning) = Number of favorable outcomes / Total number of possible outcomes
P(winning) = 1 / C(23, 5)
To calculate C(23, 5), we can use the formula:
C(n, r) = n! / (r!(n-r)!)
where n is the total number of elements and r is the number of elements to be chosen.
In this case, C(23, 5) = 23! / (5!(23-5)!)
Simplifying further:
C(23, 5) = 23! / (5! * 18!)
Finally, we can calculate the probability:
P(winning) = 1 / (23! / (5! * 18!))
P(winning) = (5! * 18!) / 23!
Note that the factorial of a number n (denoted by n!) is the product of all positive integers from 1 to n.
Therefore, to find the exact numerical value of P(winning), you can plug in these formulas into a calculator or statistical software that allows factorial calculations.