Probability Calculation - Lottery

[Hart]

Homework Statement

To win a lottery, must pick 5 different numbers from the 45 available.

The order in which the numbers are chosen does not matter.

With only one ticket, what is the probability of winning (i.e. matching all 5 numbers drawn with all 5 chosen) ?

Homework Equations

Stated within the solution

The Attempt at a Solution

n = number of elements in the field (in this case, 45)
p = number of choices (5)

$$P(win) = \left(\frac{n!}{(p!(n - p)!)}\right)$$

Therefore:

$$=\left(\frac{45!}{(5!(45- 5)!)}\right)$$


$$=\left(\frac{45!}{(5!)(40!)}\right)$$


$$=\left(\frac{45!}{(120)(40!)}\right)$$


$$=\left(\frac{45!}{(5!(40)!)}\right)$$


$$= \left(\frac{45*44*43*42*41}{120}\right)$$


$$=1221759$$

Therefore:

$$P(win)= \left(\frac{1}{1221759}\right) \approx 8.18\times10^{-7}$$

.. Is this correct method / answer?

 

[kuruman]

$$\frac{45!}{(5!)(40!)}=\frac{45*44*43*42*41*40!}{5!*40!}=\frac{45*44*43*42*41}{5!}$$

Is there a question here?

 

[Hart]

Is there a question here?

.. Just edited my initial post to make it a bit clearer and to correct some errors. Wanted to know if this was the correct method for calculating the probability?, and hence the correct answer?

 

[kuruman]

It is correct.

 

[paranormal]

I cannot confirm the accuracy of your calculation without knowing the specific lottery you are referring to and its rules. However, the method you have used is generally correct for calculating the probability of winning a lottery with 5 numbers chosen from a field of 45 numbers. It is important to note that this calculation assumes that each number has an equal chance of being drawn, which may not always be the case in real life lotteries. Additionally, this calculation only applies to a single ticket. The probability of winning increases if multiple tickets are purchased.