a certain state's lottery game, lotto, is set up so that each player pays $1 per ticket and chooses six different numbers from 1 to 49. if the six numbers chosen match the six numbers drawn each Saturday evening, the player wins the top cash prize. the order in which the numbers are drawn does not matter.how much must a person spend so that his probability of winning is 1/2
To determine how much a person must spend so that their probability of winning the lottery is 1/2, we need to calculate the number of possible combinations and understand the odds.
In the game of lotto, a player chooses 6 numbers from a pool of 1 to 49. Since the order of the numbers does not matter, we are dealing with combinations rather than permutations.
To calculate the number of possible combinations, we can use the formula for combinations, denoted as "nCr":
nCr = n! / (r!(n-r)!)
where n is the total number of items to choose from, and r is the number of items to be chosen.
In this case, we have 49 numbers to choose from, and we want to choose 6 numbers. Plugging these values into the formula:
49C6 = 49! / (6!(49-6)!)
Now, we can calculate the total number of possible combinations:
49C6 = 13,983,816
This means that there are 13,983,816 possible combinations of 6 numbers that can be chosen.
To find out how much a person must spend to have a 1/2 (50%) probability of winning, we need to determine how many tickets they would need to purchase.
Let's denote the number of tickets as T. The probability of winning with a single ticket is 1 out of the total number of possible combinations:
1 / 13,983,816
For the probability of winning to be 1/2, T tickets must be purchased such that the probability of winning is:
T / 13,983,816 = 1/2
To solve for T, we can rearrange the equation:
T = (1/2) * 13,983,816
T = 6,991,908
Therefore, a person would need to spend approximately $6,991,908 to have a 1/2 probability of winning the lotto game.