Problem 26 - College Algebra

Question

A state lottery is designed so that a player chooses six numbers from 1 to 30 on one lottery ticket. What is the probability that a player with one lottery ticket will win? What is the probability of winning if 100 different lottery tickets are purchased?

Step-by-Step Solution

Answer

The probability of winning the lottery with one ticket is approximately \(1.68 x 10^-6\) (0.00000168) or 1 in 593,775. If 100 different lottery tickets are purchased, the probability of winning increases to approximately \(1.68 x 10^-4\) (0.000168) or 100 in 593,775.

Step 1: Calculate the total number of combinations

To calculate the total number of combinations, you would use the formula for combinations: \[C(n, r) = n! / [r!(n - r)!]\] where \(n\) is the total number of items to choose from, \(r\) is the number of items to choose, and \(!\) is the factorial function. For this problem, \(n = 30\) and \(r = 6 \), so \[C(30, 6) = 30! / [6!(30 - 6)!]\] Using factorial calculations, the total number of combinations is 593,775.

Step 2: Calculate the probability of winning with one ticket

Since there is only one successful outcome (if your chosen numbers are the winning numbers), the probability of winning with just one ticket is the ratio of the successful outcomes to the total outcomes, which equals: \[P(win) = 1 / C(30, 6)\] substituting for \(C(30, 6)\) we have \[P(win) = 1 / 593775\]

Step 3: Calculate the probability of winning with 100 tickets

If 100 different tickets are purchased, that means out of the 593,775 possible combinations, we now hold 100. The probability of winning would therefore be: \[P(win) = 100 / 593,775\]